Question

Determine whether you would use factoring, square roots, or completing the square to solve the equation. Explain your reasoning. Then solve the equation. \(4 x^2-20=0\)

Short answer

\(x = \pm \sqrt{5}\)

Step-by-step solution

Rearrange the equation

Bring 20 to the other side of the equation to better understand its structure: \(4x^2 = 20\)

Divide by 4

To reduce the x term to just \(x^2\), divide both sides of the equation by 4: \(x^2 = 5\)

Find the square root

Find the square root of both sides. Don't forget that when square rooting an equation, it can be negative or positive: \(x = \pm \sqrt{5}\)

Key concepts

Factoring

Factoring is one of the primary methods to solve quadratic equations. A quadratic equation generally follows the form \(ax^2 + bx + c = 0\). Before you can use factoring, ensure that all terms are on one side of the equation. This means your equation should equal zero.

Once your equation is set to zero, your job is to express it as a product of binomials.

• The method involves finding two numbers that multiply to give \(ac\) (the product of the coefficient of \(x^2\) and the constant term \(c\)) and add up to \(b\) (the coefficient of \(x\)).
• Next, rewrite the middle term using the two numbers you found, and group the terms to factor them out.

In cases like \(4x^2-20 = 0\), you can simplify the equation first to check for any possible factorization but in this particular case, simplifying does not lead to factoring since no integer factors work. Hence, another method like using square roots might be more appropriate here.

Square Roots

The square roots method is particularly useful when the quadratic equation is missing its middle term (the \(bx\) part). This method is suitable for equations of the form \(ax^2 = c\) where the \(b\) term is zero or is absent.

In the example equation \(4x^2 - 20 = 0\), first rearrange it to \(4x^2 = 20\). Notice that it's perfect for the square root method since there is no \(b\) term to simplify.

• Start by isolating \(x^2\). You do this by dividing both sides of the equation by \(a\), simplifying it to \(x^2 = 5\).
• To find \(x\), take the square root of both sides of the equation. Remember, you should consider both the positive and negative roots, because squaring either gives the same result.

Therefore, the solution is \(x = \pm \sqrt{5}\). The square root method works perfectly here because it allows you to find solutions directly and easily.

Completing the Square

Completing the square is a powerful tool for solving quadratic equations. It's especially useful when an equation cannot be easily factored, and it provides insight into the vertex form of a quadratic. This method involves turning the quadratic into a perfect square trinomial, so it can be expressed as \((x - p)^2 = q\).

However, in the given equation \(4x^2 - 20 = 0\), we simplified to \(x^2 = 5\). There is no linear \(bx\) term to help reformat it into the \((x - p)^2\) style.

• If a true quadratic were present, you'd move the constant to the other side of the equation and add and subtract inside the square to balance it out.
• Ultimately, you're aiming to rewrite the quadratic as a binomial squared. This method then makes it possible to take the square root of both sides to solve for \(x\).

Though powerful, completing the square isn't needed for the current equation because the square root method directly solves the simplified form. But it’s an essential tool for more complex equations.