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QuestionSolve the equation 2z2+3z−1=02 z^{2}+3 z-1=0 and provide simplified answers, separated by commas.

Studdy Solution

STEP 1

Assumptions1. The equation is a quadratic equation in the form of ax+bx+c=0ax^{} + bx + c =0. . The coefficients are a =, b =3, and c = -1.
3. The solutions for the equation can be found using the quadratic formula x=−b±b−4acax = \frac{-b \pm \sqrt{b^{}-4ac}}{a}.

STEP 2

First, we need to plug the values of a, b, and c into the quadratic formula.
z=−b±b2−4ac2az = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}z=−±2−4∗2∗(−1)2∗2z = \frac{- \pm \sqrt{^{2}-4*2*(-1)}}{2*2}

STEP 3

implify the equation under the square root.
z=−3±9+8z = \frac{-3 \pm \sqrt{9+8}}{}

STEP 4

Calculate the value under the square root.
z=−3±174z = \frac{-3 \pm \sqrt{17}}{4}

STEP 5

Now, we have two possible solutions for z, one using the positive square root and one using the negative square root.
z=−3+174,−3−174z = \frac{-3 + \sqrt{17}}{4}, \frac{-3 - \sqrt{17}}{4}So, the solutions of the equation are z=−3+174z = \frac{-3 + \sqrt{17}}{4} and z=−3−174z = \frac{-3 - \sqrt{17}}{4}.

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