Problem 325 - Elementary Algebra

Question

Solve. \(x^{2}+7 x+12=0\)

Step-by-Step Solution

Answer

The solutions are \(x = -3\) and \(x = -4\).

Step 1: Identify coefficients

The given quadratic equation is in the form of \(ax^2 + bx + c = 0\). Identify the coefficients: \(a = 1\), \(b = 7\), and \(c = 12\).

Step 2: Use the quadratic formula

The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Plug in the coefficients, \(a = 1\), \(b = 7\), and \(c = 12\).

Step 3: Calculate the discriminant

Calculate the discriminant using the formula \(D = b^2 - 4ac\).Plug in the values \(b = 7\), \(a = 1\), and \(c = 12\): \(D = 7^2 - 4(1)(12)\) \(D = 49 - 48\) \(D = 1\)

Step 4: Compute the roots

Since the discriminant \(D = 1\) is positive, there are two real roots. Compute the roots using the quadratic formula: \(x = \frac{-b \pm \sqrt{D}}{2a}\).Plug in \(b = 7\), \(D = 1\), and \(a = 1\): \(x = \frac{-7 \pm \sqrt{1}}{2(1)}\) Simplify to get the two roots: \(x = \frac{-7 + 1}{2} = -3\) \(x = \frac{-7 - 1}{2} = -4\)

Step 5: Write the final solution

The two roots of the quadratic equation are \(x = -3\) and \(x = -4\).

Key Concepts

quadratic formula

To solve quadratic equations like the one given in the exercise, we often use the quadratic formula. The quadratic formula is a mathematical tool that lets us find the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are the coefficients of the equation. The formula is given by:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
This formula works by calculating the values of \(x\) that make the equation true. Remember, the symbols \(\pm\) mean that there will be two solutions: one involving addition and one involving subtraction. This is why quadratic equations usually have two roots.

discriminant

The discriminant is a part of the quadratic formula and helps us determine the nature of the roots of a quadratic equation. It is given by the expression under the square root in the quadratic formula: \(b^2 - 4ac\). The discriminant (D) can tell us if the roots are real or complex:
\[ D = b^2 - 4ac \]
The value of the discriminant helps us to understand the number and type of roots of the equation.

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), the equation has exactly one real root (also called a repeated root).
• If \(D < 0\), the equation has two complex roots.

In the given exercise, the discriminant is calculated as \(D = 1\). Since it is positive, we conclude that there are two real roots.

real roots

Real roots are the solutions to a quadratic equation that can be plotted on the real number line. When the discriminant is positive (D > 0), the quadratic equation has two distinct real roots. These roots are where the parabola formed by the equation intersects the x-axis.
For the equation \(x^2 + 7x + 12 = 0\), the steps to find real roots using the quadratic formula are already explained:

• Calculate the discriminant \(D = 1 \)
• Use the quadratic formula: \(x = \frac{-7 \pm \sqrt{1}}{2 \cdot 1}\)
• Solve to get: \(x = \frac{-7 + 1}{2} = -3\) and \(x = \frac{-7 - 1}{2} = -4\)

Therefore, the real roots of the equation are \(x = -3\) and \(x = -4\).