Solving 7x² + 3x = 2 Using The Quadratic Formula

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In mathematics, quadratic equations are polynomial equations of the second degree. They have the general form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. Solving quadratic equations involves finding the values of x that satisfy the equation, which are also known as the roots or solutions of the equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this article, we will focus on using the quadratic formula to solve a given quadratic equation and round the answer to the nearest hundredth.

Understanding the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form ax² + bx + c = 0. It provides a direct method for finding the roots of any quadratic equation, regardless of whether it can be factored easily or not. The formula is derived from the process of completing the square and is given by:

x = (-b ± √(b² - 4ac)) / 2a

Where:

  • x represents the solutions or roots of the quadratic equation.
  • a, b, and c are the coefficients of the quadratic equation in the standard form ax² + bx + c = 0.
  • The symbol ± indicates that there are generally two solutions, one with addition (+) and one with subtraction (-) in the formula.
  • The expression b² - 4ac under the square root is called the discriminant. The discriminant determines the nature of the roots:
    • If b² - 4ac > 0, the equation has two distinct real roots.
    • If b² - 4ac = 0, the equation has one real root (a repeated root).
    • If b² - 4ac < 0, the equation has two complex roots.

Steps to Apply the Quadratic Formula

Before diving into solving the specific equation, let's outline the general steps for using the quadratic formula:

  1. Identify the coefficients: Write the quadratic equation in the standard form ax² + bx + c = 0 and identify the values of a, b, and c. This is a crucial first step because the correct identification of these coefficients is essential for the accurate application of the quadratic formula. Pay close attention to the signs of the coefficients; a negative sign can significantly alter the results. For instance, in the equation 2x² - 5x + 3 = 0, a is 2, b is -5, and c is 3. Similarly, in the equation -x² + 4x - 1 = 0, a is -1, b is 4, and c is -1. A clear and correct identification of these coefficients will pave the way for the subsequent steps in solving the quadratic equation.

  2. Substitute the values into the formula: Once you have identified a, b, and c, substitute these values into the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. This step involves replacing the variables in the formula with the numerical values you've identified. It's important to perform this substitution carefully, ensuring that each value is placed correctly. Using parentheses when substituting values, especially negative ones, can help prevent errors. For example, if b is -5, substituting it into -b should be done as -(-5), which simplifies to 5. Accurate substitution is critical as it sets the foundation for the rest of the calculation. Any mistake in this step will propagate through the rest of the solution, leading to an incorrect answer.

  3. Simplify the expression: After substituting the values, simplify the expression step by step. Start by calculating the discriminant, which is the expression under the square root (b² - 4ac). This will give you a single number that you can then use to determine the nature of the roots (real, repeated, or complex). Next, simplify the rest of the expression, following the order of operations (PEMDAS/BODMAS). This may involve squaring numbers, multiplying, adding, subtracting, and finally, taking the square root. It is important to be methodical and show your work clearly to avoid making errors in arithmetic. Simplifying the expression correctly is key to arriving at the correct solutions for the quadratic equation.

  4. Calculate the two possible solutions: The ± sign in the quadratic formula indicates that there are usually two possible solutions. Calculate both solutions separately: one using the plus sign and one using the minus sign. This is where you split the formula into two separate equations: x = (-b + √(b² - 4ac)) / 2a and x = (-b - √(b² - 4ac)) / 2a. Solve each equation independently to find the two possible values of x. These values are the roots of the quadratic equation. If the discriminant (b² - 4ac) is positive, you will get two distinct real roots. If the discriminant is zero, you will get one real root (a repeated root). If the discriminant is negative, you will get two complex roots. Calculating both solutions accurately is essential to fully solve the quadratic equation.

  5. Round to the nearest hundredth (if required): If the question requires the answer to be rounded to the nearest hundredth, perform the rounding after you have calculated the solutions. Rounding should be done only at the final step to maintain accuracy. Look at the digit in the thousandths place (the third digit after the decimal point). If it is 5 or greater, round up the hundredths digit. If it is less than 5, leave the hundredths digit as it is. For example, if a solution is 2.345, it rounds up to 2.35. If a solution is 2.344, it rounds down to 2.34. Rounding to the nearest hundredth provides a practical level of precision for many applications.

Solving the Given Equation: 7x² + 3x = 2

Now, let's apply the quadratic formula to solve the given equation: 7x² + 3x = 2. Our first step is to rewrite the equation in the standard form ax² + bx + c = 0. To do this, we subtract 2 from both sides of the equation:

7x² + 3x - 2 = 0

Now we can identify the coefficients:

  • a = 7
  • b = 3
  • c = -2

Next, we substitute these values into the quadratic formula:

x = (-3 ± √(3² - 4 * 7 * -2)) / (2 * 7)

Now, we simplify the expression step by step. First, calculate the discriminant:

b² - 4ac = 3² - 4 * 7 * -2 = 9 + 56 = 65

Now, substitute the discriminant back into the formula:

x = (-3 ± √65) / 14

Next, we calculate the two possible solutions:

x₁ = (-3 + √65) / 14 x₂ = (-3 - √65) / 14

Using a calculator, we find the approximate values:

x₁ ≈ (-3 + 8.06) / 14 ≈ 5.06 / 14 ≈ 0.36 x₂ ≈ (-3 - 8.06) / 14 ≈ -11.06 / 14 ≈ -0.79

Finally, we round the solutions to the nearest hundredth:

x₁ ≈ 0.36 x₂ ≈ -0.79

Therefore, the solutions to the equation 7x² + 3x = 2, rounded to the nearest hundredth, are x = 0.36 and x = -0.79.

Conclusion

The quadratic formula is an invaluable tool for solving quadratic equations. By understanding the formula and following the steps outlined, you can solve any quadratic equation, regardless of its complexity. Remember to identify the coefficients correctly, substitute them carefully into the formula, simplify the expression step by step, and calculate both possible solutions. When required, round the solutions to the specified decimal place. In the given example, we successfully used the quadratic formula to find the solutions to the equation 7x² + 3x = 2, demonstrating the power and versatility of this mathematical tool.