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Solving Quadratic Equations

Solving Quadratic Equations

Quadratic equations are defined as equations of second degree where at least one variable or term is raised to a power of 2. Solving quadratic equations is done by determining the roots of the equation, also known as x-intercepts. These are the values of x at which the graph cuts through the x-axis, as seen below.

graph showing x intercepts of parabola StudySmarterParabola showing where the x-intercepts or roots are located

How do you solve quadratic equations?

Quadratic Equations are solved using one of the following methods:

Taking square roots

Taking square roots is a method that can be used to solve quadratic equations when there is only one term in the equation. It is done by isolating the term and then using a square root to solve the equation by finding the values of x.

Step 1: Isolate the squared variable.

Step 2: Solve your quadratic equation by calculating the square root of both sides of the equation. Remember that you will have two solutions because the square root of a number can either be positive or negative.

Factoring

Factoring is when we determine the terms that need to be multiplied together to get a mathematical expression. Factoring quadratic equations can be done in the following ways:

Taking the greatest common factor

Taking the greatest common factor (GCF) is a factoring method where we determine the highest term that evenly divides into all other terms. Let's see how this method works with an example:

Step 1: Find the greatest common factor by identifying the numbers and variables that each term has in common.

As we can see the factors that are common to both terms are 7 and x, therefore the GCF=7x.

Step 2: Write out each term as a product of the greatest common factor and another factor, i.e. the two parts of the term. The other factor can be determined by dividing your term by your GCF.

Step 3: Having rewritten each term, rewrite the quadratic equation in the following form: ab+ac=0

Step 4: Apply the law of distributive property and factor out the greatest common factor.

Step 5: Equate the factored expression to 0 and find the x-intercepts.

Perfect square

The perfect square method is when we transform a perfect square trinomial, into a perfect square binomial, . Let's have a look at solving quadratic equations using this method:

Step 1: Transform your equation from standard form, , into a perfect square trinomial, .

Step 2: Transform the perfect square trinomial into a perfect square binomial,

Step 3: Calculate the value of the x-intercept by equating the perfect square binomial to 0 and solving for x.

Grouping

Grouping is when we group terms that have common factors before factoring. Let's look at an example:

Step 1: List out the values of a, b and c.

Step 2: Find the factors that when multiplied equal , and when added equal b. T where numbers that product ac and also add to b.

The two numbers are therefore 3 and -10, as they add to -7. The other factors of 30 cannot be arranged in any way that would make them equal to -7.

Step 3: Use these factors to rewrite the x-term (bx) in the original expression/equation.

Step 4: Use grouping to factor the expression. Group the first two terms and the last two terms together, then pull out common factors from both groups and combine like terms.

Step 5: Equate the factored expression to 0 and solve for the x-intercepts.

Completing the square

Completing the square is when we change the standard form of the quadratic equation into a perfect square with an additional constant. This means changing , where m is a real number and n is a constant. They are calculated in the following way: and

Step 1: List out the values of a, b and c.

Step 2: Calculate the value of m by using the following equation:

Step 3: Calculate the value of n by using the following equation:

Step 4: Substitute your calculated values and value of a into the following equation: Step 5: Equate your equation to 0 and thereby solve the equation.

Quadratic formula

The quadratic formula is a formula that uses the coefficients and constants in a quadratic equation to solve the equation by determining its x-intercepts/roots. The quadratic formula is used to solve quadratic equations that are very difficult to factor. The quadratic formula is:

is what we refer to as the discriminant. Depending on its sign, we know how many solutions the given quadratic equation has. It is represented by the following symbol:

  • A positive discriminant means that the quadratic equation has two different real number solutions.
  • A negative discriminant means that none of the solutions are real numbers.
Real numbers are numbers that can be identified on a timeline. For example, infinity is not a real number because it doesn't have a measurable size and therefore can't be identified on a number line.
  • A discriminant that is equal to 0 means that the quadratic equation has a repeated real number solution.

Solving Quadratic Equations Discriminant graph interperatation StudySmarterGraphical representation of what the discriminant shows

The following steps will show us how to solve a quadratic equation by using the quadratic formula:

Step 1: List out the values of a, b and c.

Step 2: Calculate the value of the discriminant.

Step 3: Substitute the values of a,b and c into the Quadratic Formula and solve for both roots/solutions.

Solving Quadratic Equations - Key takeaways

  • Quadratic equations are solved by determining the roots of the equation.
  • Taking square roots is a method that can be used to solve quadratic equations when there is only one in the equation.
  • Factoring is when we determine the terms that need to be multiplied together to get a mathematical expression and can be done by taking the greatest common factor, perfect square and grouping.
  • Completing the square is when we change the standard form of the quadratic equation into a perfect square with an additional constant.
  • The quadratic formula is a method used to solve quadratic equations;

Frequently Asked Questions about Solving Quadratic Equations

Taking square roots, factoring, completing the square and using the quadratic formula

Step 1: List out the values of a,b and c. 

Step 2:  Calculate the value of m by using the following equation: m=b/2a

Step 3:  Calculate the value of n using the following equation: n=c-(b²/4a)

Step 4: Substitute your calculated values and value of a into the following equation: a(x+m)²+n

Step 5: Equate your equation to 0 and thereby solve the equation. 

Final Solving Quadratic Equations Quiz

Question

How are quadratic equations solved? 

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Answer

Quadratic equations are solved by determining its roots

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Question

What are the roots of a quadratic equation? 

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Answer

The roots of a quadratic equation are its x-intercepts. They are the points where the graph cuts through the x-axis. 

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Question

Briefly state what taking the greatest common factor is. 

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Answer

Taking the greatest common factor is when we determine the highest term that evenly divides into all other terms. 

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