Problem 47 - Algebra 1: Concepts and Skills

Question

Solve the equation by factoring. $$ 5 x^{2}-3 x-26=0 $$

Step-by-Step Solution

Answer

The roots of the quadratic equation \( 5x^{2} - 3x - 26 = 0 \) are \( x = 2.6 \) and \( x = -2 \).

Step 1: Identify the coefficients

In a quadratic equation \( ax^{2} + bx + c = 0 \), 'a' is the coefficient of \( x^{2} \), 'b' is the coefficient of 'x' and 'c' is the constant term. We can identify these as: \( a = 5 \), \( b = -3 \), \( c = -26 \).

Step 2: Substituting into the quadratic formula

Let's substitute these values into the quadratic formula, which results in: \[ x = \frac{-(-3) \pm \sqrt{(-3)^{2} - 4*5*(-26)}}{2*5} \] Now, we can simplify the expression under the square root and on the whole.

Step 3: Solving for x

After simplifying, we get: \[ x = \frac{3 \pm \sqrt{9 + 520}}{10} = \frac{3 \pm \sqrt{529}}{10} = \frac{3 \pm 23}{10} \] This gives us two possible values of x: \( x = \frac{3+23}{10} = 2.6 \) and \( x = \frac{3-23}{10} = -2 \).

Key Concepts

Factoring Quadratic Equations

When we encounter a quadratic equation, one of the most common methods to find its roots is by factoring. The process requires that we write the equation as a product of two binomials. For instance, considering the equation in the textbook exercise \( 5x^{2}-3x-26=0 \), factoring would involve looking for two numbers that multiply to give the product of the coefficient of \( x^{2} \) (in this case, 5) and the constant term (here, -26), and also add up to the coefficient of the x term (which is -3).

Unfortunately, not all quadratic equations can be factored easily, especially when the numbers are not perfect squares or do not have integer root solutions. In such scenarios, we resort to other methods like completing the square or using the quadratic formula, which is a reliable alternative to factoring for finding the roots of any quadratic equation.

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation of the form \( ax^{2} + bx + c = 0 \). It provides a straightforward method to find the roots without factoring. The formula is \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \], where 'a', 'b', and 'c' are the coefficients of the quadratic equation.

To apply the quadratic formula, follow the step-by-step solution from the exercise: first identify the coefficients (a, b, and c), then substitute them into the formula, and finally, solve for x. For the given equation \( 5x^{2}-3x-26=0 \), after substituting the identified coefficients into the quadratic formula and simplifying, we obtained the roots \( x = 2.6 \) and \( x = -2 \). This approach will always yield the correct solutions as long as the coefficients are substituted correctly and the square root is evaluated accurately.

Coefficients of a Quadratic Equation

Understanding the roles of coefficients in a quadratic equation is crucial for solving it efficiently. The standard form of a quadratic equation is \( ax^{2} + bx + c = 0 \), where 'a', 'b', and 'c' are known as the coefficients. The 'a' coefficient controls the parabola's width and direction of opening, 'b' influences the location of the vertex horizontally, and 'c' represents the y-intercept.

In our example \( 5x^{2}-3x-26=0 \), 'a' is 5, 'b' is -3, and 'c' is -26. These coefficients direct us to the most appropriate method for solving the equation—whether that is factoring, completing the square, or applying the quadratic formula. Knowing how to identify and interpret these coefficients is integral to making the solving process more efficient.