How do you solve #8m^2 - 2m = 7# using the quadratic formula?

Question

Eli Assouline · Accepted Answer

See a solution process below:

The equation should first be put in standard form: #8m^2 - 2m - color(red)(7) = 7 - color(red)(7)# #8m^2 - 2m - 7 = 0# The quadratic equation can now be used to solve this problem: According to the quadratic formula, For #color(red)(a)x^2 + color(blue)(b)x + color(green)(c) = 0#, the values of #x# which are the solutions to the equation are given by: #x = (-color(blue)(b) +- sqrt(color(blue)(b)^2 - (4color(red)(a)color(green)(c))))/(2 * color(red)(a))# Replacing: #color(red)(8)# for #color(red)(a)# #color(blue)(-2)# for #color(blue)(b)# #color(green)(-7)# for #color(green)(c)# gives: #x = (-color(blue)(-2) +- sqrt(color(blue)(-2)^2 - (4 * color(red)(8) * color(green)(-7))))/(2 * color(red)(8))# #m = (2 +- sqrt(4 - (32 * color(green)(-7))))/16# #m = (2 +- sqrt(4 - (-224)))/16# #m = (2 +- sqrt(4 + 224))/16# #m = (2 +- sqrt(228))/16# #m = (2 +- sqrt(4 * 57))/16# #m = (2 +- sqrt(4)sqrt(57))/16# #m = (2 +- 2sqrt(57))/16# #m = 2/16 +- (2sqrt(57))/16# #m = 1/8 +- sqrt(57)/8# #m = (1 +- sqrt(57))/8# The Set of Solutions Is: #m = { ((1 - sqrt(57))/8), ((1 + sqrt(57))/8)}#

Savannah Anderson · Answer

undefined To solve the equation 8m^2 - 2m = 7 using the quadratic formula, follow these steps:

1. Identify the coefficients of the quadratic equation:
   - $a = 8$
   - $b = -2$
   - $c = -7$

2. Substitute the values of $a$, $b$, and $c$ into the quadratic formula:
   $$m = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$$

3. Plug in the values:
   $$m = \frac{{-(-2) \pm \sqrt{{(-2)^2 - 4(8)(-7)}}}}{{2(8)}}$$

4. Simplify inside the square root:
   $$m = \frac{{2 \pm \sqrt{{4 + 224}}}}{{16}}$$
   $$m = \frac{{2 \pm \sqrt{{228}}}}{{16}}$$

5. Further simplify:
   $$m = \frac{{2 \pm 2\sqrt{57}}}{{16}}$$

6. Divide numerator and denominator by 2:
   $$m = \frac{{1 \pm \sqrt{57}}}{{8}}$$

Therefore, the solutions for the equation $8m^2 - 2m = 7$ using the quadratic formula are:
$$m = \frac{{1 + \sqrt{57}}}{{8}} \text{ and } m = \frac{{1 - \sqrt{57}}}{{8}}$$