Question

LOGICAL REASONING Consider the equation \(a x^{2}+b x+c=0\) and use the quadratic formula to justify the statement. If \(b^{2}-4 a c\) is zero, then the equation has one solution.

Short answer

The statement 'If \(b^{2}-4 a c\) is zero, then the equation has one solution.' is true as per the definition of discriminant in the Quadratic formula.

Step-by-step solution

Understanding the Quadratic Formula

The quadratic formula is used to solve quadratic equations and is as follows: \(x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\) Where \(a\), \(b\), and \(c\) are coefficients of the equation \(a x^{2} + b x + c = 0\), and \(x\) are the solutions to the equation. The expression under the square root, \(b^{2}-4 a c\), is called discriminant.

Conditions on the discriminant

The discriminant in the quadratic formula, \(b^{2}-4 a c\), determines the number and nature of solutions to the quadratic equation. If the discriminant is greater than zero, the equation has two real solutions; if it is equal to zero, the equation has one real solution; and if it is less than zero, the equation has two complex solution.

Evaluating The Statement

The provided statement says 'If \(b^{2}-4 a c\) is zero, then the equation has one solution.' From Step 2, we can verify that this statement is true. If the discriminant is zero, then the square root term in the quadratic formula disappears resulting in one real solution.