Question

In Exercises 41-62, solve the quadratic equation using any convenient method. \(x^{2}=64\)

Short answer

The solutions to the equation \(x^{2} = 64\) are \( x = -8 \) and \( x = 8 \).

Step-by-step solution

Recognize the Type of Equation

Recognize that the given equation \(x^{2}=64\) is a standard quadratic equation, where a = 1, b = 0, and c = -64, given in the form \(ax^{2} + bx + c = 0\).

Transpose 64 To The LHS

Rewrite the equation in standard form. Transpose 64 to the left hand side of the equation to have \(x^{2} - 64 = 0\).

Solve For X

As the equation is now in standard form, the roots of the equation can be found by setting \(x^{2} - 64 = 0\) to zero, which implies \(x^{2} = 64\), therefore x is equal to the square root of 64. Giving us two possible solutions \(x = -8 \) or \(x = 8 \)

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula itself is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula gives us the solutions, or "roots", of the equation. Here are the steps to apply it:

• Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation.
• Substitute these values into the formula.
• Calculate the discriminant \(b^2 - 4ac\), which tells us about the nature of the roots.
• Use the formula to find the values of \(x\).

The discriminant is particularly interesting as it dictates the type of solutions:- If the discriminant is positive, we have two distinct real roots.- If it's zero, there's exactly one real root.- If negative, the roots are complex.

Completing the Square

Completing the square transforms a quadratic equation into a perfect square trinomial, making it easier to solve. The process might seem intricate at first, but it's straightforward:1. Ensure the equation is in the form \(ax^2 + bx + c = 0\).2. If \(a\) is not equal to 1, divide the entire equation by \(a\).3. Move the constant term to the other side of the equation.4. Take half of the \(b\)-coefficient, square it, and add to both sides.5. Rewrite the left side as a squared binomial.6. Solve for \(x\) using square roots.This method is particularly handy if the quadratic does not factor neatly, making it a versatile option when other methods fall short.

Factoring

Factoring is a method where you express the quadratic equation in terms of two binomials. This method works best when you're dealing with equations that can be easily factored. Here's how you do it:

• Write the equation in the form \(ax^2 + bx + c = 0\).
• Find two numbers whose product equals \(a \times c\) and sum equals \(b\).
• Split the middle term (\(b\)) using the two numbers found.
• Factor by grouping: take out the common factors in the pairs, resulting in the same binomial factor appearing twice.
• Set each binomial to zero and solve for \(x\).

For example, in the equation \(x^2 - 64=0\), factoring was straightforward because \(64\) is a perfect square, allowing us to express it as \((x-8)(x+8)=0\).

Square Roots

Using square roots is one of the simplest methods to solve a quadratic equation, especially when there is no linear term. In the exercise \(x^2 = 64\), we see a perfect example.To solve:

• Recognize the equation as \(x^2 = k\).
• Take the square root of both sides of the equation, carrying over the \(\pm\) sign to account for both positive and negative values.
• For \(x^2 = 64\), we simplify to \(x = \pm 8\).

This gives us two solutions, \(x=8\) and \(x=-8\), as both satisfy the original equation. This straightforward approach is efficient when dealing with equations that are directly expressible as a perfect square.