Problem 40 - Algebra for College Students

Question

Use the quadratic formula to solve each of the quadratic equations. Check your solutions by using the sum and product relationships. $$4 x^{2}=3$$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{\sqrt{3}}{2} \) and \( x = -\frac{\sqrt{3}}{2} \).

Step 1: Rewrite the Equation in Standard Form

To use the quadratic formula, we need the equation in the standard form of a quadratic equation: \[ ax^2 + bx + c = 0 \]The given equation is:\[ 4x^2 = 3 \]Rearrange it to:\[ 4x^2 - 3 = 0 \]Here, \(a = 4\), \(b = 0\), and \(c = -3\).

Step 2: Apply the Quadratic Formula

The quadratic formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute the values of \(a\), \(b\), and \(c\): \[ x = \frac{-0 \pm \sqrt{0^2 - 4 \times 4 \times (-3)}}{2 \times 4} \]This simplifies to:\[ x = \frac{\pm \sqrt{48}}{8} \]Which simplifies further to:\[ x = \frac{\pm \sqrt{16 \times 3}}{8} \]\[ x = \frac{\pm 4\sqrt{3}}{8} \]\[ x = \frac{\pm \sqrt{3}}{2} \]So the solutions are:\( x = \frac{\sqrt{3}}{2} \) and \( x = -\frac{\sqrt{3}}{2} \).

Step 3: Check Solutions Using Sum and Product

For a quadratic equation \( ax^2 + bx + c = 0 \), the sum and product of the roots \( r_1 \) and \( r_2 \) are:\[ r_1 + r_2 = -\frac{b}{a} \]\[ r_1 \times r_2 = \frac{c}{a} \]With \( a = 4 \), \( b = 0 \), and \( c = -3 \), we have:\[ r_1 + r_2 = 0 \]\[ r_1 \times r_2 = -\frac{3}{4} \]Now check the roots:\[ \frac{\sqrt{3}}{2} + (-\frac{\sqrt{3}}{2}) = 0 \] (Sum is correct)\[ \frac{\sqrt{3}}{2} \times (-\frac{\sqrt{3}}{2}) = -\frac{3}{4} \] (Product is correct)The solutions satisfy the sum and product relationships.

Key Concepts

Quadratic Equation

Quadratic equations are fundamental in algebra and appear in the form \( ax^2 + bx + c = 0 \). These equations graph as parabolas and can have two, one, or no real solutions. Quadratic equations are significant because they model various real-world problems, such as calculating areas and projectile motion.

• In our exercise, the given equation is \( 4x^2 = 3 \).
• We first need to rewrite it in standard form: \( 4x^2 - 3 = 0 \).
• Here, \( a = 4 \), \( b = 0 \), and \( c = -3 \).

Breaking it down, we understand that the quadratic equation allows us to use various methods (like the quadratic formula) for finding the values of \( x \) that satisfy the equation. Understanding the basics of quadratic equations is the first step toward solving them efficiently.

Roots of Quadratic Equations

The roots of quadratic equations, also known as solutions or zeroes, are the values of \( x \) that make the equation zero. These roots are where the parabola intersects the x-axis. Using the quadratic formula, we can find these roots even when factoring is not straightforward.

• The formula for roots is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
• In our problem, substituting \( a = 4 \), \( b = 0 \), and \( c = -3 \), we calculate: \[ x = \frac{\pm \sqrt{48}}{8} \]
• This simplifies to \( x = \frac{\sqrt{3}}{2} \) and \( x = -\frac{\sqrt{3}}{2} \).

These solutions mean the parabola intersects the x-axis at these points, providing the values of \( x \) that solve the equation. Recognizing and calculating the roots is crucial to understanding quadratic behaviors.

Standard Form of Quadratic Equations

The standard form of a quadratic equation is expressed as \( ax^2 + bx + c = 0 \). This form is essential because it sets the stage for using formulas and methods like completing the square or using the quadratic formula to find the roots.

• From our example, \( 4x^2 = 3 \) needs to be converted to \( 4x^2 - 3 = 0 \), which is in standard form.
• Identifying \( a = 4 \), \( b = 0 \), and \( c = -3 \) is key to applying the quadratic formula.

Being in standard form helps to systematically approach solving the quadratic. This structured format makes it possible to utilize a broad range of algebraic techniques to find solutions. Learning to express equations in their standard form is critical for tackling any quadratic problem.

Sum and Product of Roots

The sum and product of roots offer a quick way to verify solutions found by other methods. For a quadratic equation \( ax^2 + bx + c = 0 \), the sum \( r_1 + r_2 \) is \( -\frac{b}{a} \), and the product \( r_1 \times r_2 \) is \( \frac{c}{a} \). This provides both a validation tool and an insight into the equation's properties.

• In our exercise, \( a = 4 \), \( b = 0 \), and \( c = -3 \).
• We calculate the sum as \( 0 \) and the product as \( -\frac{3}{4} \).
• The roots \( \frac{\sqrt{3}}{2} \) and \( -\frac{\sqrt{3}}{2} \) prove these relationships, confirming their correctness.

Understanding these relationships enables a deeper comprehension of how the roots relate back to the coefficients of the quadratic equation. This method not only confirms correctness but also underscores the interconnectedness of quadratic solutions.