Math

Question Solve the quadratic equation 0.0625x2=0.125x+1.50.0625 x^{2} = 0.125 x + 1.5 for the value of xx.

Studdy Solution

STEP 1

Assumptions
1. We are given the quadratic equation 0.0625x2=0.125x+1.50.0625 x^{2} = 0.125 x + 1.5.
2. We will solve for xx by setting the equation to zero and factoring or using the quadratic formula.

STEP 2

First, we need to set the equation to zero by moving all terms to one side of the equation.
0.0625x20.125x1.5=00.0625 x^{2} - 0.125 x - 1.5 = 0

STEP 3

To make the equation easier to work with, we can multiply all terms by 100 to remove the decimals.
100×0.0625x2100×0.125x100×1.5=100×0100 \times 0.0625 x^{2} - 100 \times 0.125 x - 100 \times 1.5 = 100 \times 0

STEP 4

Now, calculate the new coefficients.
6.25x212.5x150=06.25 x^{2} - 12.5 x - 150 = 0

STEP 5

We can try to factor the quadratic equation, but it may not factor nicely. If it does not factor easily, we will use the quadratic formula.
The quadratic formula is given by:
x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
where aa, bb, and cc are the coefficients from the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0.

STEP 6

Identify the coefficients aa, bb, and cc from the equation 6.25x212.5x150=06.25 x^{2} - 12.5 x - 150 = 0.
a=6.25a = 6.25 b=12.5b = -12.5 c=150c = -150

STEP 7

Plug the coefficients into the quadratic formula.
x=(12.5)±(12.5)24×6.25×(150)2×6.25x = \frac{-(-12.5) \pm \sqrt{(-12.5)^2 - 4 \times 6.25 \times (-150)}}{2 \times 6.25}

STEP 8

Simplify the terms inside the formula.
x=12.5±156.25+375012.5x = \frac{12.5 \pm \sqrt{156.25 + 3750}}{12.5}

STEP 9

Calculate the value inside the square root.
x=12.5±3906.2512.5x = \frac{12.5 \pm \sqrt{3906.25}}{12.5}

STEP 10

Take the square root.
x=12.5±62.512.5x = \frac{12.5 \pm 62.5}{12.5}

STEP 11

Now, solve for the two possible values of xx.
x1=12.5+62.512.5x_1 = \frac{12.5 + 62.5}{12.5} x2=12.562.512.5x_2 = \frac{12.5 - 62.5}{12.5}

STEP 12

Calculate the values for x1x_1 and x2x_2.
x1=7512.5=6x_1 = \frac{75}{12.5} = 6 x2=5012.5=4x_2 = \frac{-50}{12.5} = -4
The solutions to the equation 0.0625x2=0.125x+1.50.0625 x^{2} = 0.125 x + 1.5 are x=6x = 6 and x=4x = -4.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord