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Question Find the two values of xx that satisfy the quadratic equation 5.15x2+2.03x−1.69=05.15 x^{2} + 2.03 x - 1.69 = 0.

Studdy Solution

STEP 1

Assumptions
1. The given equation is a quadratic equation of the form ax2+bx+c=0ax^2 + bx + c = 0.
2. The coefficients of the equation are a=5.15a = 5.15, b=2.03b = 2.03, and c=−1.69c = -1.69.
3. We will use the quadratic formula to find the values of xx.
4. The quadratic formula is x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

STEP 2

Write down the quadratic formula that we will use to solve the equation.
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

STEP 3

Substitute the coefficients aa, bb, and cc from the given quadratic equation into the quadratic formula.
x=−2.03±(2.03)2−4⋅5.15⋅(−1.69)2⋅5.15x = \frac{-2.03 \pm \sqrt{(2.03)^2 - 4 \cdot 5.15 \cdot (-1.69)}}{2 \cdot 5.15}

STEP 4

Calculate the discriminant, which is the part under the square root in the quadratic formula: b2−4acb^2 - 4ac.
Discriminant=(2.03)2−4⋅5.15⋅(−1.69)\text{Discriminant} = (2.03)^2 - 4 \cdot 5.15 \cdot (-1.69)

STEP 5

Compute the value of the discriminant.
Discriminant=4.1209+34.7820=38.9029\text{Discriminant} = 4.1209 + 34.7820 = 38.9029

STEP 6

Now that we have the discriminant, we can find the square root of it to use in the quadratic formula.
Discriminant=38.9029\sqrt{\text{Discriminant}} = \sqrt{38.9029}

STEP 7

Calculate the square root of the discriminant.
Discriminant=38.9029≈6.238\sqrt{\text{Discriminant}} = \sqrt{38.9029} \approx 6.238

STEP 8

Substitute the square root of the discriminant back into the quadratic formula.
x=−2.03±6.2382⋅5.15x = \frac{-2.03 \pm 6.238}{2 \cdot 5.15}

STEP 9

Calculate the two possible values for xx using the plus and minus variations of the quadratic formula.
For the plus variation: x1=−2.03+6.2382⋅5.15x_1 = \frac{-2.03 + 6.238}{2 \cdot 5.15}
For the minus variation: x2=−2.03−6.2382⋅5.15x_2 = \frac{-2.03 - 6.238}{2 \cdot 5.15}

STEP 10

Compute the value for x1x_1.
x1=4.20810.3x_1 = \frac{4.208}{10.3}

STEP 11

Compute the value for x2x_2.
x2=−8.26810.3x_2 = \frac{-8.268}{10.3}

STEP 12

Simplify the results to find the two values of xx.
x1=4.20810.3≈0.4085x_1 = \frac{4.208}{10.3} \approx 0.4085
x2=−8.26810.3≈−0.8027x_2 = \frac{-8.268}{10.3} \approx -0.8027
The two values of the unknown xx are approximately 0.40850.4085 and −0.8027-0.8027.

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