Solving Quadratic Equations to Find a Number Whose Square Plus Twice the Number Plus One Equals 100

Imagine you are presented with a puzzle that requires you to find a number. The puzzle states that when you square the number, add twice the number, and then add one, the result should be 100. How would you solve such a puzzle? Let's break it down step by step using the mathematical tool of quadratic equations.

Setting Up the Equation

The first step is to represent the unknown number symbolically. Let's denote the number as x. According to the puzzle, the equation we need to solve is:

Step 1: Formulating the Equation

[x^2 2x 1 100]

To simplify the equation, we move all terms to one side to set it to zero:

Step 2: Simplifying and Rearranging

[x^2 2x 1 - 100 0]

This results in our simplified equation:

Step 3: Quadratic Equation Simplified

[x^2 2x - 99 0]

Solving the Quadratic Equation

Now that we have our quadratic equation, we can solve it using the quadratic formula:

Step 4: Applying the Quadratic Formula

The quadratic formula is given by:

[x frac{-b pm sqrt{b^2 - 4ac}}{2a}]

In this equation, the coefficients are:

(a 1) (b 2) (c -99)

Substituting these values into the formula, we get:

Step 5: Subulating into the Quadratic Formula

[x frac{-2 pm sqrt{2^2 - 4 cdot 1 cdot -99}}{2 cdot 1}]

[x frac{-2 pm sqrt{4 396}}{2}]

[x frac{-2 pm sqrt{400}}{2}]

[x frac{-2 pm 20}{2}]

This gives us two possible values for (x):

Step 6: Calculating the Two Possible Values

[x frac{-2 20}{2} frac{18}{2} 9]

[x frac{-2 - 20}{2} frac{-22}{2} -11]

Therefore, the two numbers that satisfy the given condition are 9 and -11.

Verifying the Solutions

To verify, let's substitute these values back into the original equation:

Verification for x 9

[9^2 2 cdot 9 1 81 18 1 100]

Verification for x -11

[(-11)^2 2 cdot (-11) 1 121 - 22 1 100]

Both values satisfy the original condition, confirming that the numbers are 9 and -11.

Conclusion

The problem of finding a number whose square plus twice the number plus one equals 100 can be solved effectively by setting up a quadratic equation and using the quadratic formula to find the roots. The steps involved are straightforward and can be applied to solve similar problems.

Related Keywords:

- Quadratic equation

- Number

- Solving equations