Problem 74 - Algebra and Trigonometry

Question

Solve: \(x(x-7)=18\)

Step-by-Step Solution

Answer

The solutions are \(x = 9\) and \(x = -2\).

Step 1: Expand the equation

First, expand the equation to make it easier to handle. The given equation is \( x(x - 7) = 18 \). Expand it to get: \[ x^2 - 7x = 18 \]

Step 2: Rearrange to form a quadratic equation

Next, rearrange the equation to form a standard quadratic equation: \( x^2 - 7x - 18 = 0 \). This is in the form \( ax^2 + bx + c = 0 \)

Step 3: Factorize the quadratic equation

Now, factorize the quadratic equation \( x^2 - 7x - 18 = 0 \). Look for two numbers that multiply to \(-18\) and add to \(-7\). These numbers are \(-9\) and \(2\). Therefore, the equation can be written as: \[ (x - 9)(x + 2) = 0 \]

Step 4: Solve for x

Finally, solve for \(x\) by setting each factor equal to zero: 1) \( x - 9 = 0 \) gives \( x = 9 \) 2) \( x + 2 = 0 \) gives \( x = -2 \)

Key Concepts

factoring quadratics

Factoring quadratics is a method used to solve quadratic equations that can be expressed as a product of two binomials. The goal is to rewrite the quadratic equation in such a way that it is easier to solve. This process involves finding two numbers that multiply together to give the constant term (in our case, \(c = -18\)) and add together to give the coefficient of the middle term (here, \(b = -7\)). For example, in the equation \(x^2 - 7x - 18 = 0\), the numbers \(-9\) and \(+2\) satisfy these conditions because \((-9) \times 2 = -18\) and \((-9) + 2 = -7\). So, the factorization of \(x^2 - 7x - 18 = 0\) is \((x - 9)(x + 2) = 0\).

Once factored, you can set each binomial equal to zero and solve for the variable \(x\):

• \(x - 9 = 0\) gives \(x = 9\)
• \(x + 2 = 0\) gives \(x = -2\)

This gives us the solutions or roots to the quadratic equation.

quadratic equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \e 0\). These equations usually have two solutions or roots.

They form a parabolic graph, and the roots of the equation are the points where the graph intersects the x-axis. To solve quadratic equations, you can use various methods like:

• Factoring: Rewriting the equation as a product of binomials
• Quadratic formula: Using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Completing the square: Rewriting the equation in the form \( a(x-h)^2+k=0\)

In the exercise, we used factoring since it was the most straightforward approach for the given quadratic equation.

roots of equations

Roots of equations, sometimes called solutions or zeros, are the values of \(x\) that satisfy the equation. For quadratic equations like \(ax^2 + bx + c = 0\), there can be up to two distinct real roots depending on the discriminant \(b^2 - 4ac\).

The three possible scenarios for the number of roots in a quadratic equation are:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root (also called a repeated root).
• If \(b^2 - 4ac < 0\), there are no real roots, only complex roots.

In our exercise, we found two real roots \(x = 9 \) and \( x = -2 \) after factoring the quadratic equation. The roots are the values of \(x\) where the equation \(x^2 - 7x - 18 = 0\) holds true.