Problem 50 - Algebra A Combined Function

Question

Solve each equation. $$ 9 x^{2}+7 x=2 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \frac{2}{9}\) and \(x = -1\).

Step 1: Move All Terms to One Side

First, we rewrite the equation by moving all terms to the left side to equal zero. The equation becomes: \[ 9x^2 + 7x - 2 = 0 \]

Step 2: Identify Coefficients for the Quadratic Formula

The quadratic equation is in the form \(ax^2 + bx + c = 0\). Here, \(a = 9\), \(b = 7\), and \(c = -2\).

Step 3: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find \(x\). Substitute \(a = 9\), \(b = 7\), and \(c = -2\) into the formula.

Step 4: Calculate the Discriminant

The discriminant \(\Delta\) is \(b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\): \[ \Delta = 7^2 - 4 \cdot 9 \cdot (-2) = 49 + 72 = 121 \]

Step 5: Solve for x Using the Calculated Discriminant

Since the discriminant is 121, a perfect square, it indicates real and rational roots. Solve for \(x\) using the quadratic formula: \[ x = \frac{-7 \pm \sqrt{121}}{18} \] This simplifies to \[ x = \frac{-7 \pm 11}{18} \]

Step 6: Find the Two Possible Solutions

Calculate the two possible solutions from step 5:- For \(x = \frac{-7 + 11}{18} = \frac{4}{18} = \frac{2}{9}\)- For \(x = \frac{-7 - 11}{18} = \frac{-18}{18} = -1\)

Key Concepts

Quadratic Formula

When faced with a quadratic equation like \(ax^2 + bx + c = 0\), the quadratic formula is your go-to tool for finding solutions. The formula is expressed as:

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

This formula is quite powerful because it can be used to find the roots of any quadratic equation, as long as you know the coefficients \(a\), \(b\), and \(c\). To use the formula effectively, simply plug in the corresponding values from your equation.

For example, in the equation \(9x^2 + 7x - 2 = 0\), we have \(a = 9\), \(b = 7\), and \(c = -2\). These values are inserted into the quadratic formula to calculate \(x\), providing us with the potential solutions.

Discriminant

The discriminant is a key component in the quadratic formula, found under the square root symbol as \(b^2 - 4ac\). It provides crucial information about the nature of the roots without needing to solve the entire equation.

Determining the discriminant can tell us:

• If \(\Delta > 0\), there are two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root or a repeated root.
• If \(\Delta < 0\), the equation has no real roots; instead, it has two complex roots.

For the equation \(9x^2 + 7x - 2 = 0\), the discriminant calculated is 121. Since 121 is a positive perfect square, it clearly indicates that the roots are real and rational.

Solving Equations

Solving quadratic equations efficiently often involves transforming the equation into a form that can easily be handled by the quadratic formula. Begin by moving all terms to one side of the equation, aiming to achieve a structure such as \(ax^2 + bx + c = 0\).

Using the given quadratic formula, plug in the identified coefficients to resolve for \(x\). For example, from our equation, once we know \(a = 9\), \(b = 7\), and \(c = -2\), substituting these into the quadratic formula allows us to solve for \(x\) as follows:

• Calculate the discriminant \(\Delta = 49 + 72 = 121\).
• Evaluate \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Simplify to find the roots.

This step-by-step approach helps in finding accurate solutions.

Real and Rational Roots

After solving the quadratic equation, you'll find the nature of its roots. Since we found that the discriminant \(\Delta = 121\) is a perfect square, the roots of this equation are real and rational.

Rational roots mean that both solutions can be expressed as fractions or whole numbers. In our given problem, the solutions turn out to be:

• \(x = \frac{2}{9}\)
• \(x = -1\)

These solutions confirm that each root is a real, rational number. This understanding helps further solidify the concepts of solving quadratic equations, encouraging comprehensive mastery of solving similar mathematical problems in the future.