Problem 98 - College Algebra

Question

Solve each quadratic equation by the method of your choice. $$ 0=-2(x-3)^{2}+8 $$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x₁ = 0.782\) and \(x₂ = 6.217\).

Step 1: Expand the equation

First, expand the squared term \(-2(x-3)^2\). This gives us: \(0 = -2(x^2 - 6x + 9) + 8\). Next, distribute to remove the brackets, which yields \(0 = -2x^2 + 12x -18 + 8\). Simplify the equation further to achieve the standard form, giving us the equation \(0 = -2x^2 + 12x -10\).

Step 2: Use the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a=-2\), \(b=12\), and \(c=-10\) into the formula, which gives us \(x = \frac{-12 \pm \sqrt{(12)^2 - 4*-2*-10}}{2*-2}\).

Step 3: Simplify and solve for x

Solve the equation \(x = \frac{-12 \pm \sqrt{144 - 40}}{-4}\). This simplifies to \(x = \frac{-12 \pm \sqrt{104}}{-4}\). Then, simplify the square root of 104 and divide each term by -4, which gives two solutions: \(x₁ = 0.782\) and \(x₂ = 6.217\).

Key Concepts

Quadratic Formula

The Quadratic Formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It allows us to find the values of \(x\) that make the equation true. The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Use this formula when other methods, like factoring, are not easily applicable.

• \(b^2 - 4ac\) is known as the discriminant. It helps determine the nature of the roots: real or complex, and whether they are distinct or repeated.
• Plug the values of \(a, b,\) and \(c\) from the quadratic equation into the formula.
• The "\(\pm\)" symbol means you will typically get two solutions, reflecting the parabolic nature of the quadratic function.

For instance, in the equation from the exercise, substituting the coefficients \(a=-2\), \(b=12\), and \(c=-10\) into the formula gives two solutions for \(x\), showing why mastering this formula is beneficial for solving any quadratic equation.

Standard Form

The Standard Form of a quadratic equation is \(ax^2 + bx + c = 0\). This is the most straightforward way to write a quadratic equation and is essential for using methods like the Quadratic Formula.
Transforming an equation into standard form often involves simplifying and rearranging terms.

• Make sure all terms are on one side of the equation.
• Organize them starting with \(x^2\)-term, followed by the \(x\)-term, and the constant term.

In our exercise, the equation was first expanded and simplified to \(0 = -2x^2 + 12x -10\). Writing it in standard form allowed easy identification of the coefficients for use in the Quadratic Formula. This form is crucial since it gives clarity regarding whether a quadratic can be factored or if the quadratic formula is necessary.

Roots of a Quadratic Equation

The roots (or solutions) of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These roots can be real numbers or complex numbers depending on the value of the discriminant \(b^2 - 4ac\). The nature of the roots provides insights into the quadratic's graph and solutions.

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root (a repeated root).
• If \(b^2 - 4ac < 0\), the roots are complex conjugates.

In the exercise provided, the discriminant is 104, which is positive, thus ensuring two distinct real roots. These roots, \(x₁ = 0.782\) and \(x₂ = 6.217\), correspond to the points where the parabola intersects the x-axis.

Completing the Square

Completing the square is another method to solve quadratic equations, transforming them into a form that can reveal the equation's roots directly. This technique writes the quadratic in the completed square form \((x - p)^2 = q\), making it easier to solve through square roots.
Here's how you complete the square:

• Start from the standard form: \(ax^2 + bx + c\).
• If \(a eq 1\), divide the entire equation by \(a\).
• Rearrange terms: place the constant term on the other side of the equation.
• Add and subtract \((\frac{b}{2})^2\) inside the equation to create a perfect square trinomial.
• Factor the trinomial into a binomial squared.

Although not used in this particular exercise, completing the square is particularly useful for understanding the properties of parabolas and can handle situations where roots involve irrational or complex numbers.