Answer:
![\large \text{$x_1$}=-\dfrac{7}{4}, \large \text{$x_2$}=\dfrac{1}{2}](https://tex.z-dn.net/?f=%5Clarge%20%5Ctext%7B%24x_1%24%7D%3D-%5Cdfrac%7B7%7D%7B4%7D%2C%20%5Clarge%20%5Ctext%7B%24x_2%24%7D%3D%5Cdfrac%7B1%7D%7B2%7D)
Step-by-step explanation:
Given: 8x² + 10x - 7 = 0
<u>When factoring trinomials of the form ax² + bx + c</u>:
- 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:
- 4x + 7 = 0
- 2x - 1 = 0
<u><em>Case 1:</em></u>
⇒ 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}}](https://tex.z-dn.net/?f=%5Cbold%7B-%5Cdfrac%7B7%7D%7B4%7D%7D)
<u><em>Case 2:</em></u>
⇒ 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}}](https://tex.z-dn.net/?f=%5Cbold%7B%5Cdfrac%7B1%7D%7B2%7D%7D)
Solutions:
![\large \text{$x_1$}=-\dfrac{7}{4}, \large \text{$x_2$}=\dfrac{1}{2}](https://tex.z-dn.net/?f=%5Clarge%20%5Ctext%7B%24x_1%24%7D%3D-%5Cdfrac%7B7%7D%7B4%7D%2C%20%5Clarge%20%5Ctext%7B%24x_2%24%7D%3D%5Cdfrac%7B1%7D%7B2%7D)
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