Question

Determine whether each value of \(x\) is a solution of the equation. Equation $$ 3 x^{2}+2 x-5=2 x^{2}-2 $$ Values (a) \(x=-3\) (b) \(x=1\) (c) \(x=4\) (d) \(x=-5\)

Short answer

Only \(x = 1\) is a solution to the equation.

Step-by-step solution

Substitution for \(x = -3\)

Substitute \(x = -3\) into the equation. This implies: \[3(-3)^{2}+2(-3)-5\] on the left side and \[2(-3)^{2}-2\] on the right side.

Evaluation for \(x = -3\)

Evaluating the expressions results in \(22\) on the left side and \(16\) on the right side hence \(x = -3\) is not a solution.

Substitution for \(x = 1\)

Substitute \(x = 1\) into the equation. This implies: \[3(1)^{2}+2(1)-5\] on the left side and \[2(1)^{2}-2\] on the right side.

Evaluation for \(x = 1\)

Evaluating the expressions results in \(0\) on both the left side and the right side hence \(x = 1\) is a solution.

Substitution for \(x = 4\)

Substitute \(x = 4\) into the equation. This implies: \[3(4)^{2}+2(4)-5\] on the left side and \[2(4)^{2}-2\] on the right side.

Evaluation for \(x = 4\)

Evaluating the expressions results in \(43\) on the left side and \(30\) on the right side hence \(x = 4\) is not a solution.

Substitution for \(x = -5\)

Substitute \(x = -5\) into the equation. This implies: \[3(-5)^{2}+2(-5)-5\] on the left side and \[2(-5)^{2}-2\] on the right side.

Evaluation for \(x = -5\)

Evaluating the expressions results in \(60\) on the left side and \(48\) on the right side hence \(x = -5\) is not a solution.

Key concepts

Solution Verification

When solving quadratic equations, it's crucial to determine whether a given value is a solution to the equation. This is known as solution verification. It helps confirm that a potential answer truly satisfies the equation.

How do we verify a solution? We substitute the proposed value into the given equation and calculate both sides separately. If both sides equal after substitution, then the value is a solution; if they do not, the value is not a solution.

For example, you have the equation:

• Left side: \(3x^2 + 2x - 5\)
• Right side: \(2x^2 - 2\)

Verification involves calculating each side using the same variable value. If both sides yield identical results, you've found a solution. It's an essential step for ensuring accuracy, particularly in complex equations, as possible mistyped or misunderstood calculations can lead us astray.

Be sure to carefully evaluate each side, as missing a simple arithmetic mistake can disrupt the verification process.

Substitution Method

The substitution method is a key technique in algebra for assessing whether a specific value solves an equation. This approach involves replacing the variable in the equation with the given number, allowing us to simplify and solve for whether the equation holds true.

Here's how you use the substitution method:

• Choose a value to test. For example, \(x = 1\).
• Substitute this value into every instance of \(x\) in the equation.
• After substitution, calculate both sides independently to see if they are equal.

This step-by-step process allows you to assess the correctness of potential solutions directly. It is methodical, reducing errors when handling more abstract or complex algebraic expressions.

The substitution method is not only used for solution verification but also assists with concepts like finding intercepts or determining graph plotting, making it widely applicable beyond simple checks.

Algebraic Expressions

Understanding algebraic expressions is crucial to mastering quadratic equations and utilizing methods like substitution effectively. An algebraic expression comprises numbers, variables, and operators (like addition and multiplication), forming the backbone of equations you'll solve.

In quadratic equations, like \(3x^2 + 2x - 5\), each term holds significance:

• \(3x^2\) is the quadratic term (with a degree of 2).
• \(2x\) is the linear term (with a degree of 1).
• \(-5\) is the constant term (a degree of 0).

These components interact through mathematical operations to define the equation. Comprehension is needed not only to substitute but to predict how changes affect the equation's balance.

Algebra requires practice in manipulating expressions, enhancing your ability to work with more complex problems over time. With a strong understanding of algebraic expressions, substitution, and solution verification becomes intuitive, leading to more accurate and confident problem-solving.