Problem 22 Use the quadratic formula to sol... [FREE SOLUTION] (2024)

Short Answer

quadratic formula

The quadratic formula is an essential tool for solving quadratic equations. A quadratic equation has the general form \(a x^{2}+b x+c=0\). The quadratic formula allows us to find the values of \(x\) that satisfy the equation. The formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It is derived from the process called completing the square.
To use the quadratic formula, simply plug in the values of \(a\), \(b\), and \(c\) from your equation and then solve for \(x\). The plus-minus sign \( \pm \) means you will often get two solutions, one using the plus, and one using the minus.

standard form

A quadratic equation must be in standard form before using the quadratic formula. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). This form ensures that the equation is ready for the next steps of solving.
For example, given the equation \(-6 a^{2}+3 a=-4\), you need to rearrange it to \(-6a^{2} + 3a + 4 = 0\).
Now it matches \(ax^2 + bx + c = 0\). Remember, the standard form is crucial as it directly provides the coefficients \(a\), \(b\), and \(c\).

discriminant

coefficients

The coefficients in a quadratic equation are the numerical constants multiplying the variables. In the standard form \(ax^2 + bx + c = 0\), the coefficients are designated as \(a\), \(b\), and \(c\).
They are essential for using the quadratic formula.
- \(a\) is the coefficient of \(x^2\).
- \(b\) is the coefficient of \(x\).
- \(c\) is the constant term.
Identifying these coefficients correctly is critical for solving equations.
For example, in \(-6a^{2} + 3a + 4 = 0\), we have \(a = -6\), \(b = 3\), and \(c = 4\).
Correctly using these in the quadratic formula ensures accurate solutions.