Solving the Number Equation
Let the unknown number be represented by the variable $x$.
Formulating the Algebraic Equation
The problem states that squaring the number ($x^2$) and subtracting twice the number ($- 2x$) results in 80 ($= 80$). This forms the equation:
$x^2 - 2x = 80$
Solving the Quadratic Equation
To find the value of $x$, we rearrange the equation into the standard quadratic form ($ax^2 + bx + c = 0$):
$x^2 - 2x - 80 = 0$
We can solve this quadratic equation by factoring. We look for two numbers that multiply to $-80$ and add to $-2$. These numbers are $-10$ and $+8$.
Factoring the equation gives:
$(x - 10)(x + 8) = 0$
This equation yields two possible solutions for $x$:
- $x - 10 = 0 \implies x = 10$
- $x + 8 = 0 \implies x = -8$
Identifying the Number from Options
The possible values for the number are 10 and -8. Reviewing the provided options (10, 15, 20, 25), the value 10 is present.
Thus, the number satisfying the condition is 10.
Important Questions from Quadratic Equation
If 2x 2+ 5x + 1 = 0, then one of the values of \(x - \frac{1}{{2x}}\) is:
If \(a-\frac{12}{a}=1\) , where a > 0, then the value of \(a^2+\frac{16}{a^2}\) is:
If x 2 – 3x + 1 = 0, then the value of \(\frac{(x^4+\frac{1}{x^2})}{(x^2+5x+1)}\) is:
If \(\sqrt{x}{}-{1\over\sqrt{x}}=\sqrt5\) , \(x \ne 0\) , then what is the value of \((x^4+{1\over{x^2}})\over(x^2+1) \) ?
If x 2 + \(\frac{1}{x^2}\) = 18, x > 0, then find the value of x 3 + \(\frac{1}{x^3}\) .
