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QuestionFind the roots of the equation f(x)=7x2+54x−16f(x)=7x^2+54x-16 using algebraic methods.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=7x+54x−16f(x)=7x^+54x-16 . We are looking for the zeros of the function, i.e., the values of xx for which f(x)=0f(x)=0

STEP 2

To find the zeros of the function, we need to solve the equation f(x)=0f(x)=0. So, we set the function equal to zero and solve for xx.
7x2+54x−16=07x^2+54x-16=0

STEP 3

This is a quadratic equation in the form ax2+bx+c=0ax^2+bx+c=0. We can solve it using the quadratic formula, which is given byx=−b±b2−ac2ax=\frac{-b\pm\sqrt{b^2-ac}}{2a}

STEP 4

In our equation, a=7a=7, b=54b=54, and c=−16c=-16. Substitute these values into the quadratic formula.
x=−54±542−4∗7∗(−16)2∗7x=\frac{-54\pm\sqrt{54^2-4*7*(-16)}}{2*7}

STEP 5

implify the expression under the square root.
x=−54±291+44814x=\frac{-54\pm\sqrt{291+448}}{14}

STEP 6

Calculate the value under the square root.
x=−54±336414x=\frac{-54\pm\sqrt{3364}}{14}

STEP 7

Take the square root.
x=−54±5814x=\frac{-54\pm58}{14}

STEP 8

Now, we have two possible solutions for xx, corresponding to the plus and minus in the formula. Calculate these two solutions.
x1=−54+5814x1=\frac{-54+58}{14}x2=−54−5814x2=\frac{-54-58}{14}

STEP 9

Calculate the values of xx and x2x2.
x=414=.2857142857142857x=\frac{4}{14}=.2857142857142857x2=−11214=−8x2=\frac{-112}{14}=-8The zeros of the function f(x)=7x2+54x−16f(x)=7x^2+54x-16 are x=.2857142857142857x=.2857142857142857 and x=−8x=-8.

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