Problem 81 - Elementary Algebra

Question

For the following problems, solve the equations by completing the square or by using the quadratic formula. $$ 5 a^{2}+5 a+4=3 a^{2}+2 a+5 $$

Step-by-Step Solution

Answer

Answer: The solutions to the given quadratic equation are $$a_{1} = \frac{-3 + \sqrt{17}}{4}$$ and $$a_{2} = \frac{-3 - \sqrt{17}}{4}$$.

Step 1: Rewrite the equation in standard form

Subtract the terms on the right side of the equation from both sides to obtain the standard form: $$5a^2 + 5a + 4 - (3a^2 + 2a + 5) = 0$$ Which simplifies to: $$2a^2 + 3a - 1 = 0$$ Now, we can identify our coefficients: $$a=2, b=3, c=-1$$.

Step 2: Apply the quadratic formula

The quadratic formula is given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Plug our values for a, b, and c into the formula: $$a = \frac{-3 \pm \sqrt{3^2 - 4\cdot 2\cdot(-1)}}{2\cdot 2}$$

Step 3: Simplify the expression

First, find the value inside the square root: $$3^2 - 4\cdot 2\cdot(-1) = 9 + 8 = 17$$ Now plug this value into the equation and do the calculation: $$a = \frac{-3 \pm \sqrt{17}}{4}$$ So, the solution for the given equation is: $$a_{1} = \frac{-3 + \sqrt{17}}{4} \quad \text{ and } \quad a_{2} = \frac{-3 - \sqrt{17}}{4}$$

Key Concepts

Completing the Square

Completing the square is a method for solving quadratic equations. It is a way to rewrite a quadratic equation in a different form to make it easier to solve. This method involves turning a quadratic expression like \( ax^2 + bx + c \) into a perfect square trinomial. This can make some problems simpler, especially when the quadratic formula might be more complex to deal with. Here's a basic outline of completing the square:

• First, make sure the equation is in the form \( ax^2 + bx + c = 0 \).
• Divide all terms by \( a \), to make the coefficient of \( x^2 \) equal to 1.
• Move the constant term \( c \) to the other side of the equation.
• Add and subtract \( \left(\frac{b}{2}\right)^2 \) on the same side of the equation to form a complete square.
• Factor the left side and solve the resulting equation by taking square roots of both sides.

This method is particularly useful when the quadratic equation cannot be factored easily.

Quadratic Formula

The quadratic formula is a universally applicable method for solving any quadratic equation of the standard form \( ax^2 + bx + c = 0 \). The formula itself is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula gives the two possible values for \( x \) (or other variables) which satisfy the equation.Here's how it works:

• Identify the coefficients \( a \), \( b \), and \( c \) from the equation.
• Plug these values into the quadratic formula.
• Simplify the expression inside the square root, known as the discriminant \( b^2 - 4ac \).
• Calculate both the addition and subtraction possibilities due to the \( \pm \) symbol.

The quadratic formula is well loved because it works with all combinations of coefficients, real or complex.

Standard Form

The standard form of a quadratic equation is crucial to solving it either by completing the square or using the quadratic formula. Standard form is given by:\[ ax^2 + bx + c = 0 \]where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \).This form is incredibly helpful because:

• It allows easy identification of coefficients needed for the quadratic formula.
• It sets up the equation perfectly for completing the square.
• All terms are on one side with zero on the other, making it straightforward to work with.

Starting with a non-standard form might require rearranging terms and simplifying the equation so that all variable terms are collected on one side of the equation.

Coefficients

The coefficients in a quadratic equation play a major role in determining the nature and number of solutions. A quadratic equation is typically written as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are the coefficients:

• \( a \): the coefficient of \( x^2 \)
• \( b \): the coefficient of \( x \)
• \( c \): the constant term

Here's why they are important:

• The value of \( a \) impacts the parabola's direction (upward if positive, downward if negative).
• \( b \) and \( a \) together determine the line of symmetry and vertex location of the parabola.
• The discriminant \( b^2 - 4ac \) indicates how many real roots exist.

Understanding coefficients helps predict the behavior of the quadratic graph and enables effective formulation of solving methods like the quadratic formula.