Consider the following quadratic equation:
8x^2−4x=1
Step 2 of 2 : Use the discriminant, b2−4ac
, to determine the number of solutions of the given quadratic equation. Then solve the quadratic equation using the formula x=−b±b2−4ac⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯/2a
To determine the number of solutions of the quadratic equation 8x^2 - 4x = 1, we need to calculate the discriminant, which is given by the formula b^2 - 4ac.
In this equation, a = 8, b = -4, and c = -1.
Discriminant = (-4)^2 - 4(8)(-1) = 16 + 32 = 48
The discriminant is positive (greater than 0), so the quadratic equation has two distinct real solutions.
To solve the quadratic equation, we will use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
Plugging in the values from the equation, we have:
x = (-(-4) ± √((-4)^2 - 4(8)(-1))) / (2(8))
= (4 ± √(16 + 32)) / 16
= (4 ± √48) / 16
Simplifying further, we have:
x = (4 ± √(16 * 3)) / 16
= (4 ± 4√3) / 16
= (1 ± √3) / 4
Therefore, the solutions to the quadratic equation 8x^2 - 4x = 1 are:
x = (1 + √3) / 4
x = (1 - √3) / 4