Problem 44
Question
\(5 a^{2}+7=852\)
Step-by-Step Solution
Verified Answer
The solutions are \(a = 13\) or \(a = -13\).
1Step 1: Isolate the term with the variable
Subtract 7 from both sides of the equation to isolate the term with the variable:\[5a^2 + 7 - 7 = 852 - 7\]This simplifies to:\[5a^2 = 845\]
2Step 2: Divide by the coefficient
Divide both sides of the equation by 5 to solve for \(a^2\):\[\frac{5a^2}{5} = \frac{845}{5}\]This simplifies to:\[a^2 = 169\]
3Step 3: Take the square root
Take the square root of both sides to solve for \(a\):\[a = \pm \sqrt{169}\]This simplifies to:\[a = \pm 13\]
Key Concepts
Isolating termsDividing by coefficientsSquare rootsAlgebraic simplification
Isolating terms
Isolating terms means moving all other numbers and variables to one side of the equation, so you are left with the term you want to solve for on the other side. Let's start with an example:
The equation is: \[5a^2 + 7 = 852\]
First, we need to isolate the term that has the variable, which is \(5a^2\). To do this, subtract 7 from both sides:
\[5a^2 + 7 - 7 = 852 - 7\]
This simplifies to:
\[5a^2 = 845\].
Now, we have \(5a^2\) by itself on one side, making it easier to solve.
The equation is: \[5a^2 + 7 = 852\]
First, we need to isolate the term that has the variable, which is \(5a^2\). To do this, subtract 7 from both sides:
\[5a^2 + 7 - 7 = 852 - 7\]
This simplifies to:
\[5a^2 = 845\].
Now, we have \(5a^2\) by itself on one side, making it easier to solve.
Dividing by coefficients
After isolating the term with the variable, the next step is to eliminate the coefficient. A coefficient is the number that is multiplied by the variable.
In our equation, the coefficient of \(a^2\) is 5. To get rid of it, divide both sides of the equation by 5:
\[\frac{5a^2}{5} = \frac{845}{5}\]
This simplifies to:
\[a^2 = 169\].
Now the equation is simpler: we have \(a^2\) by itself on one side of the equation. This makes the next step easier!
In our equation, the coefficient of \(a^2\) is 5. To get rid of it, divide both sides of the equation by 5:
\[\frac{5a^2}{5} = \frac{845}{5}\]
This simplifies to:
\[a^2 = 169\].
Now the equation is simpler: we have \(a^2\) by itself on one side of the equation. This makes the next step easier!
Square roots
After dividing by the coefficient, we need to use square roots to solve for the variable. The square root helps us find the original value of the variable before it was squared.
Take the square root of both sides of the equation:
\[a = \pm \sqrt{169}\]
This simplifies to:
\[a = \pm 13\].
Square roots always have two possible values: one positive and one negative. So in this example, \(a\) can be 13 or -13.
Take the square root of both sides of the equation:
\[a = \pm \sqrt{169}\]
This simplifies to:
\[a = \pm 13\].
Square roots always have two possible values: one positive and one negative. So in this example, \(a\) can be 13 or -13.
Algebraic simplification
In the last step, algebraic simplification helps us ensure the solution is clear and concise. Simplifying an answer means expressing it in the simplest form possible.
For our final solution, we have:
\[a = 13\] OR \[a = -13\].
There are no more steps needed here because the quadratic equation is fully solved. We clearly see that there are two possible values for the variable \(a\).
Making sure each step is fully simplified not only makes the solution easier to understand, but it also helps catch any potential errors along the way!
For our final solution, we have:
\[a = 13\] OR \[a = -13\].
There are no more steps needed here because the quadratic equation is fully solved. We clearly see that there are two possible values for the variable \(a\).
Making sure each step is fully simplified not only makes the solution easier to understand, but it also helps catch any potential errors along the way!