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Quadratic Equations
CHAPTER 2: QUADRATIC EQUATIONS
1.1 Recognize Quadratic Equations and express it in general form
General form
bx + c = 0 , where a , b and c are constants , a 0
Properties1. Equation must be in one unknown only2. The highest power of the unknown is 2
Examples1.2x 2 + 3x – 1 = 0 is a quadratic equation2.4x 2 – 9 = 0 is a quadratic equation3.8x 3 – 4x2 = 0 is not a quadratic equation
Activity 1
1. Determine whether each of the following equation is a quadratic equation or not . Equations Answer
(a) x 2 – x = 0 Yes
(b) 2x 2 – y = 0
(c) 3x + 2 = 0
(d) 2m 2 – 7m – 3 = 0
(e) k 2 – 4k = 0
(f) y 2 – 2 = 0
2. Rewrite each of the following quadratic equation in the general form. State the value of a , b and c .
Quadratic equations Value of a , b and c(a) 1 + 2x = x(x + 3) 1 + 2x = x2 + 3x x2 + x – 1 = 0
a = 1b = 1 c = -1
(b) m 2 = 21 – 4m
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Quadratic Equations
(c) (y + 6)(y – 2) = - 7
(d) x 2 =
(e) (x + 1) 2 = 16
1.2 Roots of Quadratic Equations
Notes1. The root of a quadratic equation is the value(number) of the unknown(variable) that satisfy the equation .2. A quadratic equation has at most two roots only
Exercises
1. Determine which of the values of the variable x given are roots of the respective quadratic equation.
(a) x 2 – x – 2 = 0 ; x = - 1 , 1 , 2 (b) 2x 2 + 7x + 3 = 0 ; x = - 3, - , 1 , 3
2. Determine by inspection which of the values of x are roots of the following quadratic equations .
(a) (x + 3)(x – 2) = 0 ; x = 3 , 2 , - 3 (b) x(x + 4) = 0 ; x = 4 , 0 , - 4
3. If x = 2 is the root of the quadratic equation x 2 – 3kx -10 = 0 , find the value of k .
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Quadratic Equations
2 . SOLVING QUADRATIC EQUATION
2.1 Solving Quadratic Equations
A. By Factorization If a quadratic equation can be factorized into a product of two factors such that (x – p)(x – q) = 0 , Hence x – p = 0 or x – q = 0 x = p or x = q p and q are the roots of the equation .
Notes 1. If p q the equation have two different roots
2. If p = q the equation have two equal roots (one root only)3. The equation must be written in general form ax 2 + bx + c = 0 before
factorization.
Activity 2Solve the following quadratic equations by factorization .1. x 2 – 7x – 8 = 0 ( x – 8 ) ( x + 1 ) = 0 x – 8 = 0 or x + 1 = 0 x = 8 or x = -1
2. x 2 – 4x + 4 = 0
3. x 2 – 8x = 0 4. 4x 2 – 9 = 0
5. 6x 2 + 13x – 5 = 0 6. (3x + 1)(x - 1) = 7
7. 8. (x + 1)(x – 5) = 16 x2 – 4x – 5 = 16 x2 – 4x – 21 = 0 ( x – 7 ) ( x – 3 ) = 0 x = 7 or x = 3
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Quadratic Equations
9. 10. (2p + 1)(p + 1) = 0
Exercise 1Solve the following quadratic equation by factorisation.1. x 2 – 5x – 6 = 0 [6,-1] 9. x 2 – 9x + 20 = 0 [5,4]
2. m 2 + 5m – 24 = 0 [-8,3]
10. 4x 2 – 13x + 3 = 0 [ ]
3. y 2 + 10y + 24 = 0 [-6,-4] 11. 2x 2 – 3 = 5x [ ]
4. 2x 2 + 3x – 5 = 0 [1, ] 12. 6x 2 – 11x = 7 [ ]
5. 16x 2 – 6x – 7 = 0 [ ] 13. (2x – 3) 2 = 49 [ 5,-
2]
6. 2a 2 + 4a = 0 [0.-2] 14. (3m + 1)(m – 1) = 7 [ ]
7. 100 – 9n 2 = 0 [ ] 15. 10x 2 + 4 = 13x [
]
8. (2x + 1)(x + 3 ) = 0 [ ] 16. x(x + 4) = 21 [ -7,3]
B. By Completing the Square Notes 1. The expression x 2 – 2x + 1 can be written in the form (x – 1) 2 This is called “perfect square”.
Example
Solve each of the following quadratic equation (a) (x + 1) 2 = 9 (b) x 2 = 49 x + 1 = 3 x + 1 = 3 , x + 1 = -3 x = 2 , x = - 4
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Quadratic Equations
(c) (x + 2) 2 = 36
2. From the example , note that, if the algebraic expression on the LHS of the quadratic equation are perfect squares , the roots can be easily obtained by finding the square roots.
3. To make any quadratic expression x2 + hx into a perfect square , we add the term
( to the expression .
And this will make
4. To solve the equation by using completing the square method for quadratic equation ax 2 + hx + k = 0 , follow this steps ;
Step 1 : Rewrite the equation in the form ax 2 + hx = - k Step 2 : If the coefficient of x2 is 1 , reduce the coefficient to 1 (by dividing) .
Step 3 : Add ( )2 to both sides of the equation.
Step 4 : Write the expression on the LHS as perfect square. Step 5 : Solve the equation
Examples1. x2 + 6x – 9 = 0 2. 2x2 – 5x – 8 = 0 x2 + 6x = 9
x2 + 6x + = 9 +
( x + 3 )2 = 18 x + 3 = x + 3 = 4.243 x = 4.243 – 3 , x = -4.243 – 3 x = 1.243 , x = -7.243
Exercise 2 Solve the following equations by completing the square. (Give your answers correct to four significant figures)
1. x 2 – 8x + 14 = 0 [5.41 , 2.59]
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Quadratic Equations
2. 2x 2 – 7x – 1 = 0 [3.64 , -0.14]3. x 2 + 5x + 1 = 0 [-0.209,-4.79]4. – x 2 – 3x + 5 = 0 [-4.19,1.19]5. x 2 = 5(x + 4) [7.62 , -2.62]6. -4x 2 – 12x + 3 = 0 [-3.23,0.232]7. 2x 2 – 3x – 4 = 0 [2.35,-0.85]
C. By Using the quadratic formula The quadratic equation ax 2 + bx + c can be solved by using the quadratic formula
x = , where a 0
Example 2x 2 – 7x – 3 = 0 a = 2 , b = -7 , c = -3
x = 3.886 , -0.386
Exercise 3
Use the quadratic formula to find the solutions of the following equations. Give your answers correct to three decimal places .
1. x 2 – 3x – 5 = 0 [4.193 , -1.193]2. 9x 2 = 24x – 16 [1.333 ]3. 2x 2 + 5x – 1 = 0 [0.186 , -2.686]4. 3x 2 + 14x – 9 = 0 [2.899 , -6.899]5. 7 + 5x – x 2 = 0 [0.768 , -0.434]6. m 2 = 20 – 4m [0.573 , -5.239]
7. [-1.140 , 6.140]
8. x(x + 4) = 3 [0.646 , -4.646]
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Quadratic Equations
2.2 Forming a quadratic equation from given roots
A. If the roots of a quadratic equation are known, such as x = p and x = q then, the quadratic equation is (x – p)(x – q) = 0 x 2 – px – qx + pq = 0 x 2 – (p + q)x + pq = 0
Notice that p + q = sum of roots ( SOR ) and pq = product of roots ( POR )
Hence, the quadratic equation with two given roots can be obtained as follows :-
x 2 – (SOR)x + (POR) = 0
ExamplesForm the quadratic equations from the given roots.1. x = 1 , x = 2
Method 1 Method 2 (x – 1)(x – 2) = 0 SOR = 1 + 2 = 3 x2 - 2x – x + 2 = 0 POR = 1 x 2 = 2 x2 - 3x + 2 = 0 x2 – 3x +2 = 0
2. x = - 2 , x = 3
Exercise 4
Form the quadratic equations with the given roots.
1. x = 3 , x = 2 [x2 - 5x + 6 = 0]
2. x = - 6 , [3x2 +17x - 6 = 0 ]
3. x = - 4 , x = - 6 [x2 + 10x + 24 = 0]
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Quadratic Equations
4. x = -3 , x = [5x2 + 11x - 12=0 ]
5. x = -7 , 3 [x2 + 4x - 21 = 0]
6. x = 5 only [x2 - 10x + 25 = 0]
7. x = 0 , x = [3x2 - x = 0]
8. [6x2 - 5x + 1 = 0]
B. To find the S.O.R and P.O.R from the quadratic equation in general form
ax 2 + bx + c = 0
a , x 2 + = 0
Compare with x 2 – (SOR)x + (POR) = 0
Then , SOR =
POR =
Activity 31. The roots for each of the following quadratic equations are and . Find the value of + and for the following equation
Quadratic Equationsa. x 2 – 12x + 4 = 0 12 4
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If and are the roots of the quadratic equation ax2 + bx + c = 0,
then + =
=
Quadratic Equations
b. x 2 = 4x + 8
c. 3 – 2x 2 = 10x
d. 3x 2 + 8x = 10
e. 2x 2 + 3x + 4 = 0
C. Solving problems involving SOR and POR
Activity 4
1. Given that and are the roots of the quadratic equation 2x 2 + 3x + 4 = 0 . Form a quadratic equation with roots 2 and 2.
2x 2 + 3x + 4 = 0 New roots
SOR = = 2 ( ) = = -3
= POR = 4 = 4(2) = 8
= 2 x 2 – (SOR)x + (POR) = 0 x 2 – (-3)x + 8 = 0 x 2 + 3x + 8 = 0
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Quadratic Equations
2. If and are the roots of the quadratic equation 2x 2 – 5x – 1 = 0 , form a quadratic equation with roots 3 and 3.
3. Given that and are the roots of the quadratic equation 2x 2 – 3x + 4 = 0 . Form a
quadratic equation with roots and .
4. Given that m and n are roots of the quadratic equation 2x2 – 3x – 5 = 0 , form a quadratic
equation which has the roots and .
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Quadratic Equations
Exercise 5
1. If and are roots of the quadratic equation 2x2 + 3x + 1 = 0, form a quadratic equation for the following rootsa. 2 and 2 [x2 + 3x + 2 = 0]
b. 2 + 3 and 2 + 3 [x2 - 3x + 2 = 0 ]
c. and [8x2 + 6x + 1 = 0 ]
d. 2 - 1 and 2 - 1 [x2 - 6x - 5 = 0]
2. If and are the roots of equation 2x 2 – 5x – 6 = 0 , form a quadratic
equation with roots and . [ ]
3. Given that and are the roots of the equation 3x 2 = 4 – 9x , form a quadratic equation with roots and . [ ]
4. Given m and n are the roots of the equation x 2 + 10x – 2 = 0 , form a quadratic equation with roots;
(a) 2m + 1 and 2n + 1 [ ]
(b) and [ ]
5. Given that and are the roots of the equation x 2 + 2bx + 3a = 0 , prove that 4a = b 2 .
6. Given one of the root of the quadratic equation x 2 – 5kx + k = 0 is four times the
other root, find the value of k . [ ]
7. One of the roots of the quadratic equation 2x2 + 6x = 2k – 1 is twice the value of the other root whereby k is a constant. Find the roots and the value of k.
[-1, -2 ; k = ]
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Quadratic Equations
3. DISCRIMINANT OF A QUADRATIC EQUATIONS 3.1 Determining the types of roots of quadratic equations
For the quadratic equation ax2 + bx + c = 0 , the value of b2 – 4ac will determine the types of roots.
b2 – 4ac is called the “discriminant”
Condition Type of rootsb2 – 4ac > 0 Two different rootsb2 – 4ac = 0 Two equal rootsb2 – 4ac < 0 No roots
ExampleDetermine the type of roots for each of the following quadratic equations .
(a) 2x2 – 7x + 9 = 0 (b) 2x2 – 3x – 9 = 0
a = 2 , b = -7 , c = 9 b2 – 4ac = (-7)2 – 4(2)(9) = 49 – 72 = -23 < 0 no roots
Exercise 6
Calculate the discriminant for each of the following quadratic equation and then state the type of roots for each equation .
1. x2 – 8x + 14 = 0 5. x(3x – 5) = 2x- 5
2. 2x2 – 7x – 1 = 0 6. 5(5 – 4x) = 4x2
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Quadratic Equations
3. 4 + x2 = 4x 7. x2 = 2 – 4x
4. (x – 2)2 = 3 8. 2x2 + 3x = 0
3.2 Solving problems involving the use of the discriminant
Activity 5
1. The quadratic equation 2kx2 + 4x – 3 = 0 has two equal roots , find the value of k .
2. The quadratic equation x2 + 2kx + (k + 1)2 = 0 has real roots , find the range of values of k.
PPMP Matematik TambahanNegeri Johor
Quadratic Equations
3. Show that the equation x2 + m + 1 = 8x has two different roots if m < 15 .
4. The straight line y = tx – 2 is a tangent to the graph of a curve y = 2x2 + 4x , find the value of t (t > 0) .
5. Given that the quadratic equation p(x2 + 9) = - 5qx has two equal roots , find the ratio of p : q . Hence, solve those quadratic equation .
6. Show that the quadratic equation x2 + kx = 9 – 3k has real roots for all the value of k .
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Quadratic Equations
Exercise 7
1. Find the possible values of m if the quadratic equation (4 – 2m)x2 – 2m = 1 – 3mx has two equal roots .
2. The equation x2 – 2x + = 0 has two different roots , find the range of values of k .
3. Given that the equation (p + 1)x2 – 2x + 5 = 0 has no roots , find the range of values of p .
4. Find the range value of k if the quadratic equation x2 + 1 = k – 4x has real roots .
5. The quadratic equation 2x(x – 3) = k – 2x has two distinct roots. Find the range of values of k.
6. The quadratic equation (m – 2)x 2 + 2x + 3 = 0 has two distinct roots. Find the range of values of m.
7. A quadratic equation 4x(x + 1) = 4x – 5mx – 1 has two equal roots. Find the possible values of m.
8. The straight line y = 2x – 1 does not intersect the curve y = 2x 2 + 3x + p. Find the range of values of p.
9. The straight line y = 6x + m does not intersect the curve y = 5 + 4x – x 2 . Find the range of values of m.10. The straight line y = 2x + c intersect the curve y = x2 – x + 1 at two different points, find the range of values of c.
11. Find the range values of m if the straight line y = mx + 1 does not meet the curve y2 = 4x .
12. Show that the quadratic equation kx2 + 2(x + 1) = k has real roots for all the values of k.
Answers for Exercise 7
1. 2. k < 3 3. p >
4. 5. k > -2 6. m <
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Quadratic Equations
7. m = or m = 8. p > –
9. m > 6 10. . c > 11. m > 1
Enrichment Exercise – Quadratic Equations
1. The quadratic equation kx2 + 4x + 3= 0 has two different roots, find the range of values of k .
2. Find the possible values of k if the quadratic equation x2 + (2 + k)x + 2(2 + k) = 0 has two equal roots.
3. Show that the quadratic equation x2 + (2k – 1)x + k2 = 0 has real roots if k .
4. Find the possible values of k if the straight line y = 2x + k is a tangent to the curve y = x2 + x + 1 .
5. Given that and are the roots of the quadratic equation 2x2 – 8x + 1 = 0 . Form the quadratic equation with roots and .
6. Solve each of the following quadratic equation :- a. 6x2 + 5x – 4 = 0 b. y(y + 1) = 10 c. 2x(x + 5) = 7x + 2 d. 16x2 + 8x + 1 = 0
7. The roots of the equation 2ax2 + x + 3b = 0 are and . Find the value of a and b.
8. If and are the roots of quadratic equation 2x2 – 3x – 6 = 0 , form the quadratic
equation with roots and .
9. Given and – 5 are the roots of the quadratic equation . Write the quadratic equation
in the form of ax2 + bx + c = 0 .
10. Given that m + 2 and n – 1 are the roots of the equation x2 + 5x = - 4 . Find the possible values of m and n .
11. Given that 2 and m are the roots of the equation (2x – 1)(x + 3) = k(x – 1) such that
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Quadratic Equations
k is a constant . Find the value of m and k .
12. Given one of the root of the equation 2x2 + 6x = 2k – 1 is twice the other root, such that k is a constant . Find the value of the roots and the value of k .
13. One of the root of the quadratic equation h + 2x – x2 = 0 is - 1 . Find the value of h.
14. Form the quadratic equation which has the roots -3 and . Give your answer in the
form ax2 + bx + c = 0 , where a , b and c are constants. (SPM 2004)
15. Solve the quadratic equation x(2x - 5) = 2x – 1 . Give your answer correct to three decimal places .(SPM 2005)
16. The straight line y = 5x – 1 does not intersect the curve y = 2x2 + x + p . Find the range of the values of p .(SPM 2005)
17. A quadratic equation x2 + px + 9 = 2x has two equal roots. Find the possible values of p.(SPM 2006)
Answers on Enrichment Exercises
1. k < 2. k = 6 , - 2
4. k = 5.
6. (a) (b) y = 2.702 , - 3.702 (c) (d)
7. a = 3 , b = -4 8. 9. 10. n = 0 , - 3 ; m = - 6 , - 3
11. m = 3 , k = 15 12. roots = - 1 , -2 and k =
13. h = 3 14. 2x2 + 5x – 3 = 0
15. x = 3.35 , 0.15 16. p > 1
17. p = -8 , 4
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