Question

Solve the equation by factoring. 8x^2+10x-7=0 What is the solution set?

Answer 1

Answer:

$\large x_1=-\dfrac{7}{4}, \large x_2=\dfrac{1}{2}$

Step-by-step explanation:

Given: 8x² + 10x - 7 = 0

When factoring trinomials of the form ax² + bx + c:

• multiply the leading coefficient and the last term.
• find the product factors that add to give you the coefficient of the middle term.
• rewrite the polynomial with those factors replacing the middle term.

1. Split the middle term:

⇒ 8 × -7 = -56 [multiply the leading coefficient and the last term]

⇒ 14, -4 [find the product factors that add to the middle term's coefficient]

⇒ 8x² + 14x - 4x - 7 = 0 [rewrite with those factors replacing the middle term.]

2. Factor by grouping:

⇒ 8x² + 14x - 4x - 7 = 0 [factor out 2x]

⇒ 2x(4x + 7) - 4x - 7 = 0 [factor out -1 or the negative sign]

⇒ 2x(4x + 7) -1(4x + 7) = 0 [factor out 4x + 7]

⟹ (4x + 7)(2x - 1) = 0

3. Separate into 2 cases:

1. 4x + 7 = 0
2. 2x - 1 = 0

 

Case 1:

⇒ 4x + 7 = 0 [subtract 7 from both sides]

⇒ 4x + 7 - 7 = 0 - 7

⇒ 4x = -7 [divide both sides by 4]

⇒ 4x ÷ 4 = -7 ÷ 4

⟹ x = $\bold{-\dfrac{7}{4}}$

Case 2:

⇒ 2x - 1 = 0 [subtract 1 from both sides]

⇒ 2x - 1 + 1 = 0 + 1

⇒ 2x = 1 [divide both sides by 2]

⇒ 2x ÷ 2 = 1 ÷ 2

⟹ x = $\bold{\dfrac{1}{2}}$

Solutions:

$\large x_1=-\dfrac{7}{4}, \large x_2=\dfrac{1}{2}$

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Answer 2

Answer:

$x = - \frac{7}{4} \: \frac{1}{2}$

Step-by-step explanation:

$8 {x}^{2} + 14x - 4x - 7 = 0 \\ 2x(4x + 7) - (4x + 7) = 0 \\ (4x + 7)(2x - 1) = 0$