Question

Solve the equation. Tell which solution method you used. \(x^{2}-21 x+84=0\)

Short answer

The solutions to the equation \(x^{2}-21 x+84=0\) are \(x = \frac{21 + \sqrt{105}}{2}\) and \(x = \frac{21 - \sqrt{105}}{2}\). The method used to solve this equation is the quadratic formula.

Step-by-step solution

Recognize the form

The equation \(x^{2}-21 x+84=0\) is a quadratic equation, recognized by its form \(ax^{2} + bx + c = 0\). Here, a = 1, b = -21, c = 84.

Calculate the discriminant

The discriminant of a quadratic equation \(ax^{2} + bx + c = 0\) is given by \(b^{2}-4ac\). Substituting the values, we get \((-21)^{2} - 4(1)(84) = 441 - 336 = 105.

Find the roots

The roots of the equation can be found using the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). Substituting the given values, we get: \(x = \frac{-(-21) \pm \sqrt{105}}{2(1)} = \frac{21 \pm \sqrt{105}}{2}\). This gives us two roots: \(x = \frac{21 + \sqrt{105}}{2}\) and \(x = \frac{21 - \sqrt{105}}{2}\).

Key concepts

Quadratic Formula

Understanding the quadratic formula is essential for solving equations of the second degree, which are known as quadratic equations. A quadratic equation typically takes the form of \( ax^2 + bx + c = 0 \) where \( a \) , \( b \) , and \( c \) are coefficients and \( a \) is not equal to zero.

The quadratic formula is: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). It is a powerful tool that provides the solutions to the quadratic equation by directly substituting the values of \( a \) , \( b \), and \( c \) into the formula. The formula has two parts: \( -b \) which is the negation of the second coefficient and the fraction \( \frac{1}{2a} \) that denotes halving and distributing across the two potential solutions. The \( \pm \) symbol indicates that there are usually two solutions: one involving addition and the other, subtraction.

Using the quadratic formula not only allows us to find complex solutions that may not easily factor but also gives us insights into the nature of the solutions whether they are real or complex.

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation is a component within the quadratic formula and plays a significant role in determining the nature and number of roots. It is represented by \( \Delta \) and calculated as \( b^2 - 4ac \) from the standard quadratic equation \( ax^2 + bx + c = 0 \).

The value of the discriminant can tell us three things:

• If \( \Delta > 0 \) , the equation has two distinct real roots.
• If \( \Delta = 0 \) , the equation has one real root, also known as a repeated or double root.
• If \( \Delta < 0 \) , the equation has two complex roots, which are conjugates of each other.

The discriminant thus not only aids in the calculation of the roots but also gives us a preview of the type and quantity of solutions without actually solving the equation.

Roots of a Quadratic Equation

The roots of a quadratic equation are the solutions that satisfy the equation; in other words, they are the values of \( x \) that make the quadratic expression equal to zero. They can also be referred to as zeros, x-intercepts, or solutions. For the quadratic equation \( ax^2 + bx + c = 0 \) , the roots can be found by several methods including factoring, graphing, completing the square, or using the quadratic formula.

Factoring involves finding two binomials that when multiplied together produce the original quadratic. However, it’s not always easy or possible to factor every quadratic equation, which is why the quadratic formula is such a valuable tool.

Graphing allows us to visualize where the function \( f(x) = ax^2 + bx + c \) crosses the x-axis. These points are the roots. Remember that if the roots are real, the parabola will intersect the x-axis; if the roots are complex, the parabola does not intersect the x-axis.

Each method provides the same roots, but the quadratic formula can be the most straightforward and universally applicable approach.