Question

Solve the equation. Tell which solution method you used. \(8 x^{2}+9 x-7=0\)

Short answer

The roots of the equation \(8x^{2}+9x-7=0\) are \(x_1 = \frac{-9 + \sqrt{305}}{16}\) and \(x_2= \frac{-9 - \sqrt{305}}{16}\)

Step-by-step solution

Identify a, b and c

In the quadratic equation \(ax^{2}+bx+c=0\), the coefficients are identified as \(a=8\), \(b=9\), and \(c=-7\).

Plug the coefficients into the Quadratic Formula

Substitute \(a=8\), \(b=9\), and \(c=-7\) into the Quadratic Formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) and we get \(x = \frac{-9 \pm \sqrt{(9)^2 - 4*8*(-7)}}{2*8}\)

Simplify the expression under the square root

Simplify the expression under the square root \(81 + 224 = 305\), So, \(x = \frac{-9 \pm \sqrt{305}}{16}\)

Calculate the roots

Calculate the value for both \(x_1 = \frac{-9 + \sqrt{305}}{16}\) and \(x_2= \frac{-9 - \sqrt{305}}{16}\). These are the roots of the equation.

Key concepts

Quadratic Formula

When faced with a quadratic equation like \(8x^2 + 9x - 7 = 0\), the quadratic formula is a reliable method to find the solution. This formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).

To use the quadratic formula, you first identify the coefficients in the given quadratic equation. In our example, the coefficients are \(a = 8\), \(b = 9\), and \(c = -7\). You then substitute them into the formula, which gives you the potential solutions to the equation, known as the roots. It is a method that works for any quadratic equation, whether it can be factored easily or not, making it a valuable tool for solving these types of problems.

Roots of a Quadratic

The roots of a quadratic equation are the values of \(x\) that make the equation \(ax^2 + bx + c = 0\) true. They are the points where the graph of the quadratic equation crosses the \(x\)-axis. In the quadratic formula, the roots are represented by \(x_1 = \frac{-9 + \sqrt{305}}{16}\) and \(x_2 = \frac{-9 - \sqrt{305}}{16}\) for the given problem.

The term \(\pm\) indicates that there will be two solutions: one for the addition and one for the subtraction. These solutions might be real and distinct, real and identical, or complex, depending on the discriminant \(b^2 - 4ac\). In this example, the discriminant is positive, giving us two distinct real roots.

Simplifying Expressions

Simplifying expressions is an essential step in solving quadratic equations, as it can make it easier to identify the roots. When applying the quadratic formula, simplifying the expression under the square root is crucial, as it can be the most complicated part of the formula. In our example, we simplify \(81 + 224\) to get \(305\), which is under the square root in the formula.

This process also involves simplifying the fraction by dividing the numerator by the denominator once the square root has been determined. The aim is to break down the expression to its simplest form to clearly see the possible solutions or roots for the quadratic equation.

Coefficients of a Quadratic

Quadratic equations have the standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are known as the coefficients. These coefficients play a critical role in the characteristics of the graph of the equation and in determining the roots using the quadratic formula.

In the example \(8x^2 + 9x - 7 = 0\), the coefficient \(a\) is 8, \(b\) is 9, and \(c\) is -7. The coefficient \(a\) influences the width and direction of the parabola, while \(b\) and \(c\) affect its position on the graph. Understanding these coefficients helps in visualizing the equation's graph and in predicting the nature of its roots before even performing any calculations.