Question

Determine whether the given value of \(x\) is a solution to the equation. \(x^3-6=2 x^2+9-3 x ; x=-5\)

Short answer

No, \(x = -5\) is not a solution to the given equation.

Step-by-step solution

Substitute given value of \(x\) into the equation

Firstly, the given value of \(x = -5\) is substituted into the equation \(x^3 - 6 = 2x^2 + 9 - 3x\). So the equation becomes \( (-5)^3 -6 = 2(-5)^2 + 9 - 3(-5)\).

Solve each side of the equation

Next, the calculations on both sides are performed, the left side becomes \( -5*5*5-6 = -125 - 6 = -131\). The right side becomes \( 2*5*5 + 9 + 15 = 50 + 9 + 15 = 74\).

Compare the results of both sides

Lastly, the results of the left and right side of the equation are compared. It's observed that \(-131 \neq 74\). Therefore, \(x= -5\) does not satisfy the given equation.

Key concepts

Substituting Values

The process of substituting values into an equation is a crucial step in determining whether a specific value is a solution. Let's simplify the concept. Imagine an equation as a balance scale, and the value you're substituting is like adding or removing weights to see if the scale stays balanced. Substitution requires you to replace the variable, often an 'x' or 'y', with the given number and then perform the arithmetic operations just as you would with regular numbers.

For instance, if you're given that x equals -5, you will replace every 'x' in the equation with '-5'. This means if the equation has terms like x³, you would calculate (-5)³, and for 2x², it would become 2*(-5)². This seems simple, but careful attention to arithmetic operations, especially with negative numbers and exponents, is key. Remember, substitution is the first step to verify if the value provided is indeed a solution to the equation.

Solving Equations

Moving onto the idea of solving equations, think of it as a detective's work - you're finding the value that makes the equation 'true'. After substituting the given value of the variable, your next job is to crunch the numbers. The solving step requires you to evaluate expressions on each side of the equation carefully.

Solving means simplifying the mathematical expressions by performing operations like addition, subtraction, multiplication, division, and working with exponents. In the context of our polynomial equation challenge, after substituting x = -5, we evaluated (-5)³ and 2*(-5)² along with other terms. This simplification is crucial to isolate values so we can compare them as the next logical step. If you make mistakes here, the final comparison won't hold up. Remember to follow the order of operations - parentheses, exponents, multiplication and division (from left to right), and then addition and subtraction (also from left to right).

Comparing Sides of an Equation

Finally, we reach the crucial stage of comparing sides of an equation. After the values have been substituted and the equation's sides have been solved, you must compare the results. This comparison tells you whether the substituted value keeps the equation balanced.

In the textbook example, after working out the arithmetic, we ended up with two very different numbers when we evaluated both sides of the equation with x = -5. If the resulting values from both sides are equal, then congratulations, the substituted value is a solution! If not, as in our exercise where -131 is not equal to 74, the balance is broken, and the value doesn't solve the equation. Always ensure the final comparison is between simplified results to avoid any misjudgment. This step is the definitive test and serves as the conclusive proof to validate or discredit the proposed solution.