Evaluate the discriminant for each equation. Then use it to determine the number and type of solutions. -8x² + 10x = 7

Verified step by step guidance

1. Rewrite the equation in standard quadratic form: \(-8x^2 + 10x - 7 = 0\).
2. Identify the coefficients: \(a = -8\), \(b = 10\), \(c = -7\).
3. Calculate the discriminant using the formula \(b^2 - 4ac\).
4. Substitute the values into the discriminant formula: \(10^2 - 4(-8)(-7)\).
5. Evaluate the discriminant to determine the number and type of solutions: if the discriminant is positive, there are two distinct real solutions; if zero, one real solution; if negative, two complex solutions.

Key Concepts

Discriminant

The discriminant is a key component of the quadratic formula, given by the expression b² - 4ac, where a, b, and c are coefficients from a quadratic equation in the form ax² + bx + c = 0. It helps determine the nature of the roots of the equation: if the discriminant is positive, there are two distinct real solutions; if it is zero, there is exactly one real solution; and if it is negative, there are two complex solutions.

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically expressed as ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. These equations can represent various real-world phenomena and are fundamental in algebra. To analyze them, one often rearranges the equation into standard form and applies methods such as factoring, completing the square, or using the quadratic formula.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of x that satisfy the equation. This can be achieved through various methods, including factoring, using the quadratic formula, or graphing. The solutions can be real or complex, depending on the discriminant's value, and understanding these methods is crucial for effectively evaluating and interpreting the results of quadratic equations.