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QuestionSolve the equation 5x2−7x−3=05 x^{2}-7 x-3=0 using the quadratic formula. Provide exact and approximate solutions.

Studdy Solution

STEP 1

Assumptions1. The quadratic equation is in the form of ax+bx+c=0ax^{} + bx + c =0 . The coefficients are a=5a =5, b=−7b = -7, and c=−3c = -3
3. The quadratic formula is −b±b−4aca\frac{-b \pm \sqrt{b^{} -4ac}}{a}

STEP 2

First, we need to calculate the discriminant, which is the part under the square root in the quadratic formula. The discriminant is given by b2−4acb^{2} -4ac.
Discriminant=b2−4acDiscriminant = b^{2} -4ac

STEP 3

Now, plug in the given values for the coefficients a, b, and c to calculate the discriminant.
Discriminant=(−7)2−∗5∗(−3)Discriminant = (-7)^{2} -*5*(-3)

STEP 4

Calculate the discriminant.
Discriminant=49−(−60)=109Discriminant =49 - (-60) =109

STEP 5

Now that we have the discriminant, we can plug it into the quadratic formula to find the solutions for x.
x=−b±Discriminant2ax = \frac{-b \pm \sqrt{Discriminant}}{2a}

STEP 6

Plug in the values for b, the discriminant, and a into the quadratic formula.
x=−(−)±1092∗5x = \frac{-(-) \pm \sqrt{109}}{2*5}

STEP 7

Calculate the exact solutions for x.
x=7±10910x = \frac{7 \pm \sqrt{109}}{10}These are the exact solutions for the equation.

STEP 8

Now, let's calculate the approximate solutions for x. We can do this by approximating the square root of the discriminant.
109≈10.44\sqrt{109} \approx10.44

STEP 9

Plug in the approximate value for the square root of the discriminant into the quadratic formula.
x≈7±.44x \approx \frac{7 \pm.44}{}

STEP 10

Calculate the approximate solutions for x.
x≈7+10.4410,7−10.4410x \approx \frac{7 +10.44}{10}, \frac{7 -10.44}{10}x≈.744,−0.344x \approx.744, -0.344These are the approximate solutions for the equation.
The exact solutions are x=7±10910x = \frac{7 \pm \sqrt{109}}{10} and the approximate solutions are x≈.744,−0.344x \approx.744, -0.344.

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