Solve the following quadratic equations by factoring.
Problem 1 :
x^{2} + 7x + 12 = 0
Problem 2 :
x^{2} - 9x + 20 = 0
Problem 3 :
x^{2} – 5x – 24 = 0
Problem 4 :
3x^{2} – 5x – 12 = 0
Problem 5 :
(x + 4)^{2} = 2x + 88
Problem 6 :
√5x^{2} + 2x – 3√5 = 0
Problem 7 :
x - ¹⁸⁄ₓ = 3
Problem 8 :
x^{2} - 100 = 0
Problem 9 :
9x^{2} - 64 = 0
Problem 10 :
(3x + 2)^{2} - 81 = 0
1. Answer :
In the quadratic equation x^{2} + 5x + 6 = 0, multiply the coefficient of x^{2} is 1 and the consterm 12.
= 1 x 12
= 12
Find two factors of 12 such that the product of them is equal to 12 and the sum is equal to the coeffient of x, 7.
3 x 4 = 12
3 + 4 = 7
So, the two factors of 12 are 3 and 4.
Split the middle term 7using the two factors 3 and 4.
x^{2} + 3x + 4x + 12 = 0
Factor and solve the above quadratic equation by grouping.
(x^{2} + 3x) + (4x + 12) = 0
x(x + 3) + 4(x + 3) = 0
(x + 3)(x + 4) = 0
x + 3 = 0 or x + 4 = 0
x = -3 or x = -4
2. Answer :
In the quadratic expression x^{2} - 9x + 20 = 0, multiply the coefficient of x^{2} is 1 and the consterm 20.
= 1 x 20
= 20
Find two factors of 20 such that the product of them is equal to 20 and the sum is equal to the coeffient of x, -9.
(-4) x (-5) = 20
(-4) + (-5) = -9
So, the two factors of 20 are -4 and -5.
Split the middle term 10x using the two factors -4 and -5.
x^{2} - 4x - 5x + 20 = 0
Factor and solve the above quadratic equation by grouping.
(x^{2} - 4x) + (-5x + 20) = 0
x(x - 4) - 5(x - 4) = 0
(x - 4)(x - 5) = 0
x - 4 = 0 or x - 5 = 0
x = 4 or x = 5
3. Answer :
In the quadratic expression x^{2} – 5x – 24 = 0, multiply the coefficient of x^{2}, 1 and the consterm -24.
= 1 x (-24)
= -24
Find two factors of -24 such that the product of them is equal to -24 and the sum is equal to the coeffient of x, -5.
-8 x 3 = -24
-8 + 3 = -5
So, the two factors of -24 are -8 and 3.
Split the middle term -5x using the two factors -8 and 3.
x^{2} - 8x + 3x - 24 = 0
Factor and solve the above quadratic equation by grouping.
(x^{2} - 8x) + (3x - 24) = 0
x(x - 8) + 3(x - 8) = 0
(x - 8)(x + 3) = 0
x - 8 = 0 or x + 3 = 0
x = 8 or x = -3
4. Answer :
In the quadratic expression 3x^{2} – 5x – 12 = 0, multiply the coefficient of x^{2}, 3 and the consterm -12.
= 3 x (-12)
= -36
Find two factors of -36 such that the product of them is equal to -36 and the sum is equal to the coeffient of x, -5.
-9 x 4 = -36
-9 + 4 = -5
So, the two factors of -36 are -9 and 4.
Split the middle term -5x using the two factors -9 and 4.
3x^{2} - 9x + 4x - 36 = 0
Factor and solve the above quadratic equation by grouping.
(3x^{2} - 9x) + (4x - 36) = 0
3x(x - 9) + 4(x - 9) = 0
(x - 9)(3x + 4) = 0
x - 9 = 0 or 3x + 4 = 0
x = 9 or x = -⁴⁄₃
5. Answer :
The quadratic expression (x + 4)^{2} = 2x + 88 is not in standard form. Write it in standard form.
(x + 4)^{2} = 2x + 88
(x + 4)(x + 4) = 2x + 88
x^{2} + 4x + 4x + 16 = 2x + 88
x^{2} + 8x + 16 = 2x + 88
Subtract 2x and 88 from both sides.
x^{2} + 6x - 72 = 0
In the quadratic equation x^{2} + 6x - 72 = 0, multiply the coefficient of x^{2}, 1 and the constant term -72.
= 1 x (-72)
= -72
Find two factors of -72 such that the product of them is equal to -72 and the sum is equal to the coeffient of x, 6.
-6 x 12 = -72
-6 + 12 = 6
So, the two factors of -72 are -6 and 12.
Split the middle term 6x using the two factors -6 and 12.
x^{2} - 6x + 12x - 72 = 0
Factor and solve the above quadratic equation by grouping.
(x^{2} - 6x) + (12x - 72) = 0
x(x - 6) + 12(x - 6) = 0
(x - 6)(x + 12) = 0
x - 6 = 0 or x + 12 = 0
x = 6 or x = -12
6. Answer :
In the quadratic expression √5x^{2} + 2x – 3√5 = 0, multiply the coefficient of x^{2}, √5 and the consterm -3√5.
= √5 x (-3√5)
= -15
Find two factors of -15 such that the product of them is equal to -15 and the sum is equal to the coeffient of x, 2.
-3 x 5 = -15
-3 + 5 = 2
So, the two factors of -15 are -3 and 5.
Split the middle term 2x using the two factors -3 and 5.
√5x^{2} - 3x + 5x – 3√5 = 0
Factor and solve the above quadratic equation by grouping.
(√5x^{2} - 3x) + (5x – 3√5) = 0
(√5x^{2} - 3) + (√5√5x - 3√5) = 0
x(√5x - 3) + √5(√5x - 3) = 0
(√5x - 3)(x + √5) = 0
x - 3 = 0 or x + √5 = 0
x = ³⁄√₅ or x = -√5
7. Answer :
The quadratic expression x - ¹⁸⁄ₓ = 3 is not in standard form. Write it in standard form.
x - ¹⁸⁄ₓ = 3
Multiply both sides by x.
x^{2} - 18 = 3x
Subtract 3x from both sides.
x^{2} - 3x - 18 = 0
In the quadratic expression x^{2} - 3x - 18 = 0, multiply the coefficient of x^{2}, 1 and the consterm -18.
= 1 x (-18)
= -18
Find two factors of -18 such that the product of them is equal to -18 and the sum is equal to the coeffient of x, -3.
-6 x 3 = -18
-6 + 3 = -3
So, the two factors of -18 are -6 and 3.
Split the middle term -3x using the two factors -6 and 3.
x^{2} - 6x + 3x - 18 = 0
Factor and solve the above quadratic equation by grouping.
(x^{2} - 6x) + (3x - 18) = 0
x(x - 6) + 3(x - 6) = 0
(x - 6)(x + 3) = 0
x - 6 = 0 or x + 3 = 0
x = 6 or x = -3
8. Answer :
The quadratic equation x^{2} - 100 = 0 can be solved using the following algebraic identity.
a^{2} - b^{2} = (a + b)(a - b)
x^{2} - 100 = 0
x^{2} - 10^{2} = 0
(x + 10)(x - 10) = 0
x + 10 = 0 or x - 10 = 0
x = -10 or x = 10
9. Answer :
9x^{2} - 64 = 0
3^{2}x^{2} – 8^{2 }= 0
(3x)^{2 }– 8^{2 }= 0
(3x + 8)(3x - 8) = 0
3x + 8 = 0 or 3x - 8 = 0
x = -⁸⁄₃ or x = ⁸⁄₃
10. Answer :
(3x + 2)^{2} - 81 = 0
(3x + 2)^{2} – 9^{2 }= 0
(3x + 2 + 9)(3x + 2 - 9) = 0
(3x + 11)(3x - 7) = 0
3x + 11 = 0 or 3x - 7 = 0
x = -¹¹⁄₃ or x = ⁷⁄₃
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