Problem 117 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use any convenient method to solve the quadratic equation. $$4 x^{2}-17=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(4 x^{2}-17 = 0\) are \(x = \frac{\pm \sqrt{17}}{2}\)

Step 1: Rewrite the Equation

Make \(4 x^{2}\) the subject of the formula by adding 17 to both sides which gives us \(4x^{2}=17\)

Step 2: Isolate \(x^{2}\)

Divide both sides of the equation by 4 to isolate \(x^{2}\). Now the equation looks like \(x^{2}=\frac{17}{4}\)

Step 3: Solve for \(x\)

Finally, take the square root of both sides of the equation to solve for \(x\). Don't forget that \(x\) could be both negative and positive when \(x^{2}\) is squared. Therefore, \(x\) can either be \(\sqrt{\frac{17}{4}}\) or \(-\sqrt{\frac{17}{4}}\) which gives us \(x = \frac{\pm \sqrt{17}}{2}\)

Key Concepts

Quadratic Formula

When faced with a quadratic equation in the form of \( ax^{2} + bx + c = 0 \) where \( a \) , \( b \) , and \( c \) are constants, and \( a \) is not equal to zero, one reliable method to find the roots is using the quadratic formula. The formula is \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \). It's designed to produce the two solutions, which could be real or complex, that satisfy the equation.

These solutions are found by substituting the values of \( a \) , \( b \) , and \( c \) into the formula. The term \( b^{2} - 4ac \) is known as the discriminant, which can tell us the nature of the roots; real and distinct, real and same, or complex. The square root of the discriminant divided by \( 2a \) gives us the two possible values for \( x \) - one with a plus sign and one with a minus. The reason why we have a \( \pm \) is because both \( \sqrt{x^{2}} \) is \( x \) and \( -x \).

Using the quadratic formula is a powerful technique, especially when other methods, such as factoring or completing the square, are not easily applicable.

Isolate Variable

Isolating the variable is a fundamental skill in algebra and is crucial for solving equations, including quadratic equations. The objective is to get the variable by itself on one side of the equal sign. To do this, you'll perform inverse operations to reverse the operations affecting the variable.

Let's look at an example: If you start with \( 4x^{2} - 17 = 0 \) and you want to isolate \( x ^{2} \), you first need to eliminate the constant term by adding 17 to both sides. Now you have \( 4x^{2} = 17 \). The next step is to reverse the multiplication by 4, so you divide each side of the equation by 4, resulting in \( x^{2} = \frac{17}{4} \).

Isolation of the variable is typically one of the initial steps in solving any algebraic equation. It simplifies the problem and paves the way to find the solution by using additional methods, such as taking square roots or applying the quadratic formula.

Square Root Method

The square root method can be a quick way to solve quadratic equations, but it's most effective when the equation has no linear term, i.e., \( bx = 0 \). This method involves taking the square root of both sides of an equation after isolating the \( x^{2} \) term.

In our example, \( x^{2} = \frac{17}{4} \), the next step is simply taking the square root of both sides. Remember, whenever you take the square root of both sides, you must consider both the positive and negative solutions because squaring either a positive or negative number results in a positive value.

So, the solutions will look like this \( x = \pm \sqrt{\frac{17}{4}} \), which simplifies to \( x = \frac{\pm \sqrt{17}}{2} \). The square root method can be efficient and direct, saving time and effort, especially apt for equations that are already in a form amenable to this technique.