Question

Use the quadratic formula to solve the equation. $$-3 y^{2}+2 y+8=0$$

Short answer

The roots of the equation -3y²+2y+8=0 are \( y1 = -1.333 \) and \( y2 = 2 \)

Step-by-step solution

Identify the coefficients

The quadratic equation in standard form is defined as ax²+bx+c=0. Here, a=-3, b=2, and c=8.

Apply the Quadratic Formula

The quadratic formula is given by \( \frac {-b ± \sqrt {b^2 - 4ac}}{2a} \). Here, plug a=-3, b=2, and c=8 into the quadratic formula.

Calculate the Discriminant

First, calculate the value under the square root in the quadratic formula, called the discriminant: \( b^2 - 4ac = (2)^2 - 4*(-3)*8 = 4 + 96 = 100 \)

Calculation of roots

Now take the square root of the discriminant (which is 100), and continue solving the equation: \( \frac {-b ± \sqrt {b^2 - 4ac}}{2a} = \frac {-2 ± \sqrt {100}}{2*(-3)} = \frac {-2 ± 10}{-6} \). This gives us two solutions: \( y1 = \frac {8}{-6} = -1.333 \) and \( y2 = \frac {-12}{-6} = 2 \)

Key concepts

Solving Quadratic Equations

Understanding how to solve a quadratic equation is essential for students delving into algebra. Quadratic equations are typically in the form of \( ax^2 + bx + c = 0 \), where 'a', 'b', and 'c' represent known numbers, with 'a' not equal to zero. The method you choose to solve a quadratic equation often depends on the particular equation at hand.

Using the Quadratic Formula

One of the most reliable methods is the quadratic formula: \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \). This formula allows you to find the solutions, or 'roots', of any quadratic equation if you know the coefficients 'a', 'b', and 'c'. It's a universal method because it always works, regardless of whether the roots are real or complex numbers.

When applying the formula, it's critical to substitute the values correctly and to carry out the discriminant calculation accurately, as these steps determine the nature and the exact values of the roots of the equation.

Discriminant Calculation

The discriminant is a vital component in solving quadratic equations, as it indicates the nature of the equation's roots before you even calculate them. It is the part under the square root in the quadratic formula, expressed as \( D = b^2 - 4ac \).

Determining the Nature of Roots

• If \( D > 0 \), the quadratic equation has two distinct real roots.
• If \( D = 0 \), there is exactly one real root, also known as a repeated or double root.
• If \( D < 0 \), the roots are complex and occur in conjugate pairs.

To calculate the discriminant, simply square the 'b' coefficient and subtract four times the product of 'a' and 'c'. This step is crucial to understanding what kind of solutions you should expect and informs you whether or not the roots will be real numbers. Take care to calculate the discriminant correctly to avoid errors in the subsequent steps.

Quadratic Equation Roots

After calculating the discriminant, finding the roots of a quadratic equation is the final step. The roots can be thought of as the x-values where the parabolic graph of the quadratic equation intersects the x-axis. They are the solutions to the equation \( ax^2 + bx + c = 0 \).

Real Roots

If the discriminant is non-negative, you can find the roots by applying the plus or minus part of the formula: \( {-b \pm \sqrt{D}} \). The roots are calculated by adding and subtracting the square root of the discriminant, then dividing by \( 2a \).

For the given exercise, where \( a = -3 \), \( b = 2 \), and \( c = 8 \), the quadratic formula yields two solutions, showing that this particular equation has two real roots. The values are \( y1 = -1.333 \) and \( y2 = 2 \). Students should remember that a negative discriminant would point towards complex roots, which also come in conjugate pairs.