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QuestionSolve the equation 4w2+w−7=04 w^{2}+w-7=0 using the quadratic formula. List solutions, separated by commas.

Studdy Solution

STEP 1

Assumptions1. The given equation is a quadratic equation of the form ax+bx+c=0ax^ + bx + c =0. . The quadratic formula is given by x=−b±b−4acax = \frac{-b \pm \sqrt{b^ -4ac}}{a}.
3. The solutions to the quadratic equation are real if the discriminant b−4acb^ -4ac is greater than or equal to0.

STEP 2

First, we identify the coefficients aa, bb, and cc from the given equation 4w2+w−7=04w^2 + w -7 =0.
a=4,b=1,c=−7a =4, b =1, c = -7

STEP 3

Next, we substitute the values of aa, bb, and cc into the quadratic formula.
w=−b±b2−ac2aw = \frac{-b \pm \sqrt{b^2 -ac}}{2a}w=−1±12−∗∗(−7)2∗w = \frac{-1 \pm \sqrt{1^2 -**(-7)}}{2*}

STEP 4

implify the expression under the square root (the discriminant).
w=−1±1+1128w = \frac{-1 \pm \sqrt{1 +112}}{8}

STEP 5

Calculate the value under the square root.
w=−1±1138w = \frac{-1 \pm \sqrt{113}}{8}

STEP 6

Finally, we calculate the two possible values for ww.
w=−1+1138,−1−1138w = \frac{-1 + \sqrt{113}}{8}, \frac{-1 - \sqrt{113}}{8}The solutions to the equation 4w2+w−=04w^2 + w - =0 are w=−1+1138w = \frac{-1 + \sqrt{113}}{8} and w=−1−1138w = \frac{-1 - \sqrt{113}}{8}.

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