Question
Solve by the quadratic formula: \( -8 x^{2}-10 x-3=0 \) List your answers separated by commas, for example: \( \frac{2}{3}, \frac{5}{2} \) type " \( 2 / 3,5 / 2 \) " or \( \frac{5 \pm 2 \sqrt{3}}{5} \), type " \( (5+-2 \operatorname{sqrt}(3)) / 5^{\prime \prime} \). \( \square \) Next Question
Ask by Peters Reid. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = -0.75, -0.5 \)
Solution
Solve the equation \( -8x^{2}-10x-3=0 \).
Solve the quadratic equation by following steps:
- step0: Solve by factoring:
\(-8x^{2}-10x-3=0\)
- step1: Factor the expression:
\(-\left(2x+1\right)\left(4x+3\right)=0\)
- step2: Divide the terms:
\(\left(2x+1\right)\left(4x+3\right)=0\)
- step3: Separate into possible cases:
\(\begin{align}&2x+1=0\\&4x+3=0\end{align}\)
- step4: Solve the equation:
\(\begin{align}&x=-\frac{1}{2}\\&x=-\frac{3}{4}\end{align}\)
- step5: Rewrite:
\(x_{1}=-\frac{3}{4},x_{2}=-\frac{1}{2}\)
The solutions to the quadratic equation \( -8x^{2}-10x-3=0 \) are \( x_{1}=-\frac{3}{4} \) and \( x_{2}=-\frac{1}{2} \) or \( x_{1}=-0.75 \) and \( x_{2}=-0.5 \).
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Simplify this solution The Deep Dive
First, let's rewrite the equation in the standard form by multiplying through by -1, giving us \( 8x^2 + 10x + 3 = 0 \). Now, using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): 1. Here, \( a = 8 \), \( b = 10 \), and \( c = 3 \). 2. Calculate the discriminant: \( b^2 - 4ac = 10^2 - 4 \cdot 8 \cdot 3 = 100 - 96 = 4 \). Now substitute into the quadratic formula: \[ x = \frac{-10 \pm \sqrt{4}}{2 \cdot 8} = \frac{-10 \pm 2}{16}. \] This gives us two potential answers: \[ x = \frac{-10 + 2}{16} = \frac{-8}{16} = -\frac{1}{2}, \] \[ x = \frac{-10 - 2}{16} = \frac{-12}{16} = -\frac{3}{4}. \] So the solutions are: \(-\frac{1}{2}, -\frac{3}{4}\)
