Problem 68 - Algebra 1

Question

Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. $$7 x^{2}-63=0$$

Step-by-Step Solution

Answer

The solutions for the equation are \(x=3\) and \(x=-3\).

Step 1: Rewriting the Equation

First, rewrite the equation in the standard form for a quadratic equation \(ax^{2}=k\). This gives the equation \(7x^{2}=63\)

Step 2: Isolating x^{2}

Next, isolate \(x^{2}\) by dividing both sides of the equation by 7: \(\frac{7x^{2}}{7}=\frac{63}{7}\). This simplifies the equation to \(x^{2}=9\)

Step 3: Solving For x

Now, solve for 'x'. Remember that when you take the square root of both sides, you need to consider both the positive and negative root. This gives two solutions: \(x=\sqrt{9}\) and \(x=-\sqrt{9}\). Thus, \(x=3\) and \(x=-3\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of quadratic equations, which have the general form \( ax^2 + bx + c = 0 \). With this formula, you can solve any quadratic equation, even if it doesn’t factor nicely. It is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). The term \( \sqrt{b^2 - 4ac} \) is called the discriminant and it determines the nature of the roots:

• If it’s positive, there are two real and distinct roots.
• If it’s zero, there is exactly one real root, often called a repeated root.
• If it’s negative, the roots are complex or imaginary, indicating no real solution.

For the equation \( 7x^2 - 63 = 0 \), we first notice that it simplifies to \( 7x^2 = 63 \), illustrating a case where the quadratic formula isn't necessary because the equation is set up for easy simplification.

Solving Equations

Solving equations involves finding values for the variable that make the equation true. With quadratic equations like \( ax^2 = c \), you often isolate the variable by using inverse operations.In this exercise, the equation \( 7x^2 = 63 \) is simplified by dividing each side by 7, leading us to \( x^2 = 9 \). This highlights a crucial step – isolating the \( x^2 \) term. Once isolated, we can see more clearly how to find \( x \).

• When solving \( x^2 = 9 \), you apply the principle of square roots.
• You consider both the positive and negative roots, a unique property whenever dealing with squares.

It’s essential to remember to check both roots because they will both satisfy the original squared equation. Hence, for this example, the solutions are \( x = 3 \) and \( x = -3 \). This mirrors the policy to verify solutions in algebra where both roots balance the equation.

Square Roots

Square roots are mathematical functions that undo the process of squaring a number. If you have \( x^2 = 9 \), applying the square root function to both sides gives you \( x = \sqrt{9} \). Similarly, you should also account for \( x = -\sqrt{9} \) since both values will satisfy \( x^2 \).The concept of square roots is key in solving equations that feature a squared term:

• The primary root, \( \sqrt{n} \), is the positive root.
• The secondary root, \( -\sqrt{n} \), is just as valid and vital.

In the case of \( x^2 = 9 \), both \( 3 \) and \( -3 \) are square roots, ensuring the equation holds true. Understanding this principle allows us to solve quadratic equations and other algebraic equations involving power terms. Keep in mind that when you solve these problems, observing the context can also help determine if both roots apply.