Question
What are the roots of the equation? Answer \( 2 x^{2}-3 x-2=0 \) \( \frac{-3 \pm \sqrt{7}}{4} \) \( \frac{3 \pm \sqrt{7}}{4} \) and \( \frac{1}{2} \) and -2
Ask by Chavez Cook. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The roots are \( x = -\frac{1}{2} \) and \( x = 2 \).
Solution
Solve the equation \( 2x^2-3x-2=0 \).
Solve the quadratic equation by following steps:
- step0: Solve by factoring:
\(2x^{2}-3x-2=0\)
- step1: Factor the expression:
\(\left(x-2\right)\left(2x+1\right)=0\)
- step2: Separate into possible cases:
\(\begin{align}&x-2=0\\&2x+1=0\end{align}\)
- step3: Solve the equation:
\(\begin{align}&x=2\\&x=-\frac{1}{2}\end{align}\)
- step4: Rewrite:
\(x_{1}=-\frac{1}{2},x_{2}=2\)
The roots of the equation \(2x^{2}-3x-2=0\) are \(x = -\frac{1}{2}\) and \(x = 2\).
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Simplify this solution Beyond the Answer
To find the roots of the equation \( 2 x^{2}-3 x-2=0 \), you can utilize the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For this equation, \( a = 2 \), \( b = -3 \), and \( c = -2 \). Plugging these values in, you’ll calculate the discriminant (the part under the square root), which assists in determining the roots. Once you simplify using the quadratic formula, you'll see that the roots can be expressed as \( \frac{3 \pm \sqrt{25}}{4} \), clearly leading you to \( x = 2 \) or \( x = -\frac{1}{2} \) as the correct roots. Keep an eye out for signs and arithmetic mistakes, as these can lead to confusion in your final answer!
