Problem 64 - Algebra 1

Question

The solution of a quadratic equation can be found by graphing each side separately and locating the points of intersection. You may wish to consult page 532 for help in approximating solutions. $$ -4.87 x^{2}+1.44 x=5.22 x^{2}+6 x $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = -0.27\) and \(x = 0.27\)

Step 1: Convert equation to standard quadratic form

Combine like terms on one side of the equation to convert it into the standard quadratic form. Given equation is \(-4.87x^2 + 1.44x = 5.22x^2 + 6x\). Combining like terms, we get \(-4.87x^2 - 5.22x^2 + 1.44x - 6x = 0\). This simplifies to \(-10.09x^2 - 4.56x = 0\).

Step 2: Apply the quadratic formula

Now we can apply the quadratic formula to solve the equation. The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Here, \(a = -10.09\), \(b = -4.56\), and \(c = 0\). Plugging these values into the formula, we get \(x = \frac{4.56 \pm \sqrt{(-4.56)^2 - 4*-10.09*0}}{2*-10.09}\). Simplifying, we get \(x = \frac{4.56 \pm \sqrt{20.7936}}{-20.18}\).

Step 3: Evaluate the solution

Evaluate the expression inside the square root and calculate the two values of \(x\). This results in two possible solutions for \(x\): \(x1 = \frac{4.56 + \sqrt{20.7936}}{-20.18}\) and \(x2 = \frac{4.56 - \sqrt{20.7936}}{-20.18}\). Solving these equations for \(x\), we find that \(x1 = -0.27\) and \(x2 = 0.27\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of a quadratic equation. A quadratic equation is typically in the form of \( ax^2 + bx + c = 0 \). The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula allows you to calculate the roots of the quadratic equation directly. The expression under the square root, \( b^2 - 4ac \), is known as the discriminant. It determines the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is one real root.
• If the discriminant is negative, the roots are complex.

Using the quadratic formula effectively requires identifying the coefficients \( a \), \( b \), and \( c \) from the quadratic equation and substituting them into the formula.

Standard Quadratic Form

A quadratic equation should ideally be in the standard quadratic form before you can use methods like factoring, completing the square, or applying the quadratic formula. The standard form of a quadratic equation is: \[ ax^2 + bx + c = 0 \]Here, \( a \), \( b \), and \( c \) are known as the coefficients.

• \( a \) represents the coefficient of \( x^2 \),
• \( b \) is the coefficient of \( x \), and
• \( c \) is the constant term.

To convert any quadratic equation into its standard form, you should make sure there are no terms on the right side of the equation. This usually involves combining like terms on one side and setting the equation to zero. This step is crucial as it allows for proper application of the quadratic formula or other solving methods.

Graphing Quadratic Functions

Graphing a quadratic function can help visualize its behavior, including its roots, vertex, and axis of symmetry. A typical quadratic function is represented as \( y = ax^2 + bx + c \). Here’s how to understand its graph:

• The graph of a quadratic function is a parabola.
• If \( a \) is positive, the parabola opens upwards; if negative, it opens downwards.
• The vertex is the highest or lowest point, depending on the direction of the parabola.
• The axis of symmetry is a vertical line that passes through the vertex, given by \( x = -\frac{b}{2a} \).

To find the roots of the equation visually, you can graph each side of the original equation separately and locate their intersection points. These intersection points correspond to the solution of the quadratic equation.

Solving Quadratic Equations

Solving quadratic equations means finding the values of \( x \) that make the equation true. This can be done using several methods:

• The Quadratic Formula: Use when the quadratic equation is complicated or doesn't factor easily.
• Factoring: Find factors that multiply to \( ac \) and add up to \( b \).
• Completing the Square: Rewrite the equation by making perfect square trinomial on one side.
• Graphing: Plot the equation and find the roots where the graph intersects the x-axis.

Choosing the best method often depends on the specific equation and the coefficients it contains. The quadratic formula is often a go-to because it works with any quadratic equation in standard form.