Problem 11 - Algebra for College Students

Question

Solve each equation by using the quadratic formula. $$y^{2}+y=6$$

Step-by-Step Solution

Answer

The solutions are y = 2 and y = -3.

Step 1: Write the equation in standard quadratic form

Rearrange the given equation so that one side equals zero. The given equation is \[y^2 + y = 6\] Subtract 6 from both sides to get: \[y^2 + y - 6 = 0\]

Step 2: Identify the coefficients

Identify the coefficients a, b, and c in the quadratic equation format \[a y^2 + by + c = 0\] For the equation \[y^2 + y - 6 = 0\] you have: \[a = 1, \, b = 1, \, c = -6\]

Step 3: Apply the quadratic formula

The quadratic formula is given by: \[y = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\] Substitute the values of a, b, and c into the formula: \[y = \frac{{-(1) \pm \sqrt{{(1)^2 - 4(1)(-6)}}}}{{2(1)}} = \frac{{-1 \pm \sqrt{{1 + 24}}}}{{2}} = \frac{{-1 \pm \sqrt{25}}}{{2}}\]

Step 4: Simplify under the square root

Simplify the expression under the square root: \[\sqrt{25} = 5\] Substitute back into the formula: \[y = \frac{{-1 \pm 5}}{{2}}\]

Step 5: Solve for the possible values of y

Calculate the two potential solutions by solving the plus-minus: \[y = \frac{{-1 + 5}}{{2}} = \frac{4}{2} = 2\] \[y = \frac{{-1 - 5}}{{2}} = \frac{-6}{2} = -3\]

Step 6: State the solutions

The solutions to the quadratic equation \[y^2 + y - 6 = 0\] are \[y = 2\] and \[y = -3\]

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation of the form \[ax^2 + bx + c = 0\], where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The highest power of the variable is 2, which is why it's called 'quadratic'.
Examples of quadratic equations are:

• \(3x^2 - 4x + 2 = 0\)
• \(x^2 - 7x + 10 = 0\)
• \(2y^2 + 5y - 3 = 0\)

Quadratic equations can be used to model various real-world situations, like the path of a projectile or the area of a shape.
They can be solved using various methods, including factoring, completing the square, and the quadratic formula.

Solving Equations

Solving a quadratic equation involves finding the values of the variable that make the equation true.
One of the most powerful and widely used methods to solve quadratic equations is the quadratic formula.
The quadratic formula states that for any quadratic equation of the form \[ax^2 + bx + c = 0\], the solutions for \(x\) can be found using: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\].
Here's a step-by-step approach to using the formula:

• First, identify the coefficients \(a\), \(b\), and \(c\) from the equation.
• Next, substitute these values into the formula.
• Simplify the expression under the square root (also known as the discriminant).
• Finally, solve for the values of \(x\) using the plus-minus symbol (\(\pm\)).

This process yields two solutions because of the \(\pm\) sign in the formula, which covers both the positive and negative roots.

Standard Form

The standard form of a quadratic equation is widely used because it allows for easy application of the quadratic formula.
When a quadratic equation is not in standard form, you should rearrange it so one side equals zero.
For example, if you start with an equation like \[y^2 + y = 6\], you'll rearrange it by subtracting 6 from both sides to get:\[y^2 + y - 6 = 0\].
In this format, your coefficients \(a\), \(b\), and \(c\) are clear, making it straightforward to use the quadratic formula.
The beauty of the standard form lies in its simplicity and consistency, making it much easier to apply various solving techniques.