Question

Solve the equation using any method. Explain your reasoning. \(7(x-4)^2-18=10\)

Short answer

The solutions for the equation \(7(x-4)^2-18=10\) are \(x = 6\) and \(x = 2\).

Step-by-step solution

Rewrite the Equation

The first step is to rewrite the equation \(7(x-4)^2-18=10\). You should expand the equation to its standard form. So, this results in \(7x^2 - 56x + 112 - 18 = 10\).

Simplify the Equation

Now, continue to simplify the equation by combining like terms which gives \(7x^2 - 56x + 94 = 10\). Then, subtract 10 from both sides resulting into the equation \(7x^2 - 56x + 84 = 0\). Further simplify this equation by dividing it by 7 giving \(x^2 - 8x + 12 = 0\).

Use the Quadratic Formula

After simplifying, use the quadratic formula to solve for x. The formula is \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). For our equation, a=1, b=-8, and c=12. Substituting a, b and c into the equation gives \(x = \frac{8 \pm \sqrt{(-8)^2-4*1*12}}{2*1} = \frac{8 \pm \sqrt{64-48}}{2} = \frac{8 \pm \sqrt{16}}{2}\). This further simplification gives \(x = \frac{8 \pm 4}{2}\).

Evaluate for x

Evaluating the equation gives the two solutions as \(x = \frac{8+4}{2} = 6\) and \(x = \frac{8-4}{2} = 2\). The solutions to the equation are \(6\) and \(2\).

Key concepts

Quadratic Formula

When tackling quadratic equations, the quadratic formula is a powerful tool that helps you find the solutions or "roots" of the equation. This formula is essential for solving equations of the form \(ax^2 + bx + c = 0\). The formula itself is:

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

Here is how it works:

• a: the coefficient of \(x^2\)
• b: the coefficient of \(x\)
• c: the constant term

The expression under the square root sign, \(b^2 - 4ac\), is called the "discriminant." It determines the number of real solutions:

• If the discriminant is positive, there are two real and distinct solutions.
• If it is zero, there is exactly one real solution.
• If it is negative, there are no real solutions, only imaginary ones.

For our specific equation from the exercise, we identified \(a=1\), \(b=-8\), and \(c=12\). Substituting these into the formula, we calculated the solutions as \(x = 2\) and \(x = 6\).

Standard Form of a Quadratic

The standard form of a quadratic equation is crucial for using methods like factoring, completing the square, or applying the quadratic formula. It is expressed as \(ax^2 + bx + c = 0\).

To convert an equation into this form, you may need to do some algebraic manipulation. For instance, you should combine like terms and ensure one side of the equation equals zero.

In our exercise, the given equation \(7(x-4)^2-18=10\) needed conversion. After expanding \(7(x-4)^2\) and rearranging, we obtained \(7x^2 - 56x + 84 = 0\). This is the standard form, making it ready for further solutions, such as using the quadratic formula.

Simplifying Equations

Simplifying an equation is an important step before solving it. It involves combining like terms and reducing the equation to its simplest form. This makes complicated equations easier to handle, especially when preparing to apply the quadratic formula or other solving techniques.

In the original problem, we first expanded and combined terms to get \(7x^2 - 56x + 84 = 0\). We further simplified by dividing every term by 7 to reach \(x^2 - 8x + 12 = 0\). This step is vital as it reduces the complexity of the coefficients, making calculations more straightforward. By simplifying before solving, you can minimize the potential for errors and make the problem more approachable.