Problem 50 - Contemporary Precalculus

Question

Solve the equation by any method. $$5 x^{2}+2 x=2$$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation 5x² + 2x = 2 using the quadratic formula. Answer: x = (-2 + √44)/10, (-2 - √44)/10

Step 1: Rewrite the equation in the standard quadratic form

Subtract 2 from both sides of the equation to have it in the form of $$ax^2 + bx + c = 0$$ $$ 5x^2 + 2x - 2 = 0 $$

Step 2: Identify a, b, and c

From the standard quadratic form, $$ax^2 + bx + c = 0$$, identify the values of a, b, and c: a = 5 b = 2 c = -2

Step 3: Use the quadratic formula to solve for x

The quadratic formula is given by: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Step 4: Substitute the values of a, b, and c into the formula

Plug the values of a, b, and c into the quadratic formula and simplify: $$ x = \frac{-(2) \pm \sqrt{(2)^2 - 4(5)(-2)}}{2(5)} $$

Step 5: Simplify the equation

After plugging the values, let's simplify the expression under the square root first: $$ x = \frac{-2 \pm \sqrt{4 + 40}}{10} $$ Now, simplify the square root and the final expression: $$ x = \frac{-2 \pm \sqrt{44}}{10} $$

Step 6: Write the final solution

The final solution for the quadratic equation is: $$ x = \frac{-2 + \sqrt{44}}{10}, \; \frac{-2 - \sqrt{44}}{10} $$

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations in the form of \( ax^2 + bx + c = 0 \). This formula provides a straightforward way to find the solutions, or roots, of the equation. The quadratic formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation. The term \( b^2 - 4ac \) is known as the discriminant and plays an important role in determining the nature of the roots. Plus or minus (\( \pm \)) indicates that there can be two possible solutions, depending on the sign of the square root. This formula works for every quadratic equation, even when there are complex solutions.

Standard Form of a Quadratic Equation

Any quadratic equation can be rearranged to the standard form, which is expressed as \( ax^2 + bx + c = 0 \). This form is crucial for easily identifying the coefficients \( a \), \( b \), and \( c \), which are used in the quadratic formula. Rewriting the equation in standard form often involves moving all terms to one side of the equation so that the equation equals zero. For example, the equation \( 5x^2 + 2x = 2 \) must be rearranged by subtracting 2 from both sides, resulting in \( 5x^2 + 2x - 2 = 0 \). This rearranged equation is then ready for further analysis and solving.

Solving Equations

Solving quadratic equations involves finding the values of \( x \) that satisfy the equation. After converting a quadratic equation to its standard form \( ax^2 + bx + c = 0 \), a common method of solving is using the quadratic formula. This method is preferred for its general applicability to all quadratics:

• Start by identifying the coefficients \( a \), \( b \), and \( c \).
• Substitute these values into the quadratic formula.
• Simplify the expression inside the square root (the discriminant).
• Compute the final solutions for \( x \) using the plus or minus options offered by the \( \pm \) in the formula.

This procedure ensures finding all possible roots for the equation, even if they involve irrational numbers.

Discriminant in Quadratic Equations

The discriminant is the part of the quadratic formula under the square root: \( \sqrt{b^2 - 4ac} \). It reveals important information about the nature and number of roots for the quadratic equation:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root, sometimes called a repeated or double root.
• If \( b^2 - 4ac < 0 \), the equation has two complex roots, which means no real number solutions exist.

In our example, substituting the coefficients into the discriminant gives \( 4 + 40 = 44 \), indicating that the quadratic equation has two distinct real roots. Understanding the discriminant helps in predicting the type of solutions before fully solving the equation.