Math

Question Find the roots of the quadratic equation 3x25x+13x^2 - 5x + 1 using the quadratic formula. Round the answers to the nearest hundredth if necessary, and separate multiple answers with a comma.

Studdy Solution

STEP 1

Assumptions
1. The given quadratic expression is 3x25x+13x^2 - 5x + 1.
2. We are to find the roots of the quadratic equation using the quadratic formula.
3. The quadratic formula is given by x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} for a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0.
4. The roots should be rounded to the nearest hundredth if necessary.

STEP 2

Identify the coefficients aa, bb, and cc from the quadratic expression 3x25x+13x^2 - 5x + 1.
a=3,b=5,c=1a = 3, b = -5, c = 1

STEP 3

Plug the coefficients aa, bb, and cc into the quadratic formula.
x=(5)±(5)243123x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}

STEP 4

Simplify the expression inside the square root (the discriminant).
Δ=(5)2431\Delta = (-5)^2 - 4 \cdot 3 \cdot 1

STEP 5

Calculate the discriminant Δ\Delta.
Δ=2512\Delta = 25 - 12

STEP 6

Simplify the discriminant.
Δ=13\Delta = 13

STEP 7

Now, substitute the discriminant back into the quadratic formula.
x=5±136x = \frac{5 \pm \sqrt{13}}{6}

STEP 8

Calculate the two possible values for xx by using the plus and minus signs in the quadratic formula.
x1=5+136x_1 = \frac{5 + \sqrt{13}}{6} x2=5136x_2 = \frac{5 - \sqrt{13}}{6}

STEP 9

Calculate the value of x1x_1 using a calculator or by hand.
x1=5+1365+3.616x_1 = \frac{5 + \sqrt{13}}{6} \approx \frac{5 + 3.61}{6}

STEP 10

Simplify the value of x1x_1.
x18.616x_1 \approx \frac{8.61}{6}

STEP 11

Divide to find the decimal value of x1x_1.
x11.435x_1 \approx 1.435

STEP 12

Round x1x_1 to the nearest hundredth.
x11.44x_1 \approx 1.44

STEP 13

Calculate the value of x2x_2 using a calculator or by hand.
x2=513653.616x_2 = \frac{5 - \sqrt{13}}{6} \approx \frac{5 - 3.61}{6}

STEP 14

Simplify the value of x2x_2.
x21.396x_2 \approx \frac{1.39}{6}

STEP 15

Divide to find the decimal value of x2x_2.
x20.231666...x_2 \approx 0.231666...

STEP 16

Round x2x_2 to the nearest hundredth.
x20.23x_2 \approx 0.23
The roots of the quadratic equation 3x25x+13x^2 - 5x + 1 are approximately x1=1.44x_1 = 1.44 and x2=0.23x_2 = 0.23.

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