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Solve the equation by factoring. \( 8 x^{2}+10 x-7=0 \) Rewrite the equation in factored form. \( \square=0 \) (Factor completely.) The solution set is \( \{\square\} \). (Use a comma to separate answers as needed. Type each solution only once.)

Ask by Black Weber. in the United States
Feb 03,2025

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Tutor-Verified Answer

Answer

The equation \(8x^{2}+10x-7=0\) factors to \((4x-1)(2x+7)=0\), so the solutions are \(x=-\frac{7}{4}\) and \(x=\frac{1}{2}\).

Solution

Solve the equation \( 8x^{2}+10x-7=0 \). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(8x^{2}+10x-7=0\) - step1: Factor the expression: \(\left(2x-1\right)\left(4x+7\right)=0\) - step2: Separate into possible cases: \(\begin{align}&2x-1=0\\&4x+7=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=\frac{1}{2}\\&x=-\frac{7}{4}\end{align}\) - step4: Rewrite: \(x_{1}=-\frac{7}{4},x_{2}=\frac{1}{2}\) The equation \(8x^{2}+10x-7=0\) can be factored as \((4x-1)(2x+7)=0\). Therefore, the solutions to the equation are \(x=-\frac{7}{4}\) and \(x=\frac{1}{2}\).

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Bonus Knowledge

To solve \( 8x^2 + 10x - 7 = 0 \) by factoring, we start by finding two numbers that multiply to \( 8 \times -7 = -56 \) and add to \( 10 \). These numbers are \( 14 \) and \( -4 \). Now, we rewrite the middle term: \( 8x^2 + 14x - 4x - 7 = 0 \) Next, we group the terms: \( (8x^2 + 14x) + (-4x - 7) = 0 \) Factoring gives us: \( 2x(4x + 7) - 1(4x + 7) = 0 \) Factoring out \( (4x + 7) \) results in: \( (2x - 1)(4x + 7) = 0 \) Setting each factor to zero: \( 2x - 1 = 0 \) or \( 4x + 7 = 0 \) Thus, we find the solutions: \( x = \frac{1}{2} \) or \( x = -\frac{7}{4} \) In factored form: \( (2x - 1)(4x + 7) = 0 \) The solution set is \( \{\frac{1}{2}, -\frac{7}{4}\} \).

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