Problem 10 - Algebra for College Students

Question

Solve each equation by using the quadratic formula. $$x^{2}+4 x+3=0$$

Step-by-Step Solution

Answer

The solutions are \(x = -1\) and \(x = -3\).

Step 1: Identify the coefficients

Identify the coefficients from the quadratic equation in the form \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = 4\), and \(c = 3\).

Step 2: Write the quadratic formula

Recollect the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Step 3: Substitute the coefficients

Substitute \(a = 1\), \(b = 4\), and \(c = 3\) into the quadratic formula: \[ x = \frac{-(4) \pm \sqrt{(4)^2 - 4(1)(3)}}{2(1)} \]

Step 4: Simplify inside the square root

Simplify the expression under the square root: \[ (4)^2 - 4(1)(3) = 16 - 12 = 4 \] So, the equation becomes: \[ x = \frac{-4 \pm \sqrt{4}}{2} \]

Step 5: Simplify the square root

Calculate the square root: \(\sqrt{4} = 2\). Thus, the equation is: \[ x = \frac{-4 \pm 2}{2} \]

Step 6: Calculate the solutions

Solve for \(x\) using both the positive and negative values: \[ x = \frac{-4 + 2}{2} = \frac{-2}{2} = -1 \] and \[ x = \frac{-4 - 2}{2} = \frac{-6}{2} = -3 \] So, the solutions are \(x = -1\) and \(x = -3\).

Key Concepts

Solving Quadratic Equations

When we need to solve quadratic equations, a great tool to use is the quadratic formula.
The formula is: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation in the form \(ax^2 + bx + c = 0\).
The quadratic formula helps us find the values of \(x\) for which the equation equals zero.
The key steps in solving a quadratic equation with this formula include:

• Identifying the coefficients \(a\), \(b\), and \(c\).
• Substituting these values into the quadratic formula.
• Simplifying the expression under the square root (also called the discriminant).
• Simplifying the square root.
• Finally, calculating the possible values of \(\frac{-b + \text{sqrt}(...)}{2a}\) and \(\frac{-b - \text{sqrt}(...)}{2a}\).

For the quadratic equation \(x^2 + 4x + 3 = 0\), these steps help us to find the solutions \(x = -1\) and \(x = -3\).
You also need to apply good arithmetic practices at each step to ensure accuracy.

Coefficients in Quadratic Equations

Coefficients are the numbers in front of the variables in any algebraic expression.
In a quadratic equation like \(ax^2 + bx + c = 0\), \(a\), \(b\), and \(c\) are coefficients.

• \(a\) is the coefficient of \(x^2\). It affects the width and direction of the parabola when graphed. If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards. For \(x^2 + 4x + 3 = 0\), \(a\) is 1.
• \(b\) is the coefficient of \(x\). It influences the slope of the parabola, shifting it left or right. Here, \(b\) is 4.
• \(c\) is the constant term. It affects the location where the parabola intersects the y-axis. In our example, \(c\) is 3.

Correctly identifying these coefficients paves the way for accurate solutions using the quadratic formula.

Simplifying Square Roots

Simplifying square roots is an important step when using the quadratic formula.
This process helps to make equations easier to understand and solve.
Here’s what you need to know:

• The Discriminant: The expression inside the square root \(b^2 - 4ac\) is called the discriminant. It tells us about the nature of the roots. For \(x^2 + 4x + 3 = 0\), the discriminant is \(4^2 - 4(1)(3) = 4\).
• Calculating the Square Root: Once the discriminant is simplified to \(4\), the square root of \(4\) is \(2\).
• Result Interpretation: A positive discriminant means two real solutions, zero gives one real solution, and a negative discriminant means no real solution (complex roots).

So, in our example, after simplifying \(\text{sqrt}(4)\), we further simplify the equation to get the roots \(x = -1\) and \(x = -3\).