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Question Find the roots of the quadratic equation f(x)=5x2+6x+1f(x) = 5x^2 + 6x + 1.

Studdy Solution

STEP 1

Assumptions
1. The function given is f(x)=5x2+6x+1 f(x) = 5x^2 + 6x + 1
2. We are looking for the zeros of the function, which are the values of x x that make f(x)=0 f(x) = 0

STEP 2

To find the zeros of the quadratic function, we can use the quadratic formula:
x=−b±b2−4ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
where a a , b b , and c c are the coefficients from the quadratic equation ax2+bx+c=0 ax^2 + bx + c = 0 .

STEP 3

Identify the coefficients a a , b b , and c c from the given function.
a=5,b=6,c=1 a = 5, \quad b = 6, \quad c = 1

STEP 4

Substitute the coefficients into the quadratic formula.
x=−6±62−4⋅5⋅12⋅5 x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 5 \cdot 1}}{2 \cdot 5}

STEP 5

Simplify the expression inside the square root.
x=−6±36−2010 x = \frac{-6 \pm \sqrt{36 - 20}}{10}

STEP 6

Calculate the value inside the square root.
x=−6±1610 x = \frac{-6 \pm \sqrt{16}}{10}

STEP 7

Take the square root of 16.
x=−6±410 x = \frac{-6 \pm 4}{10}

STEP 8

Now, solve for the two possible values of x x .
For the positive square root:
x=−6+410 x = \frac{-6 + 4}{10}

STEP 9

Calculate the value for the positive square root.
x=−210 x = \frac{-2}{10}

STEP 10

Simplify the fraction.
x=−15 x = -\frac{1}{5}

STEP 11

Now, solve for the value of x x using the negative square root:
x=−6−410 x = \frac{-6 - 4}{10}

STEP 12

Calculate the value for the negative square root.
x=−1010 x = \frac{-10}{10}

STEP 13

Simplify the fraction.
x=−1 x = -1
The zeros of the function f(x)=5x2+6x+1 f(x) = 5x^2 + 6x + 1 are x=−15 x = -\frac{1}{5} and x=−1 x = -1 .

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