Evaluating Algebraic Expressions with Radicals
We are asked to find the value of the expression \(x^2-4x-10\) given the value of \(x\). The value of \(x\) is given as:
\(x = 2 + 2^{1/2} + 2^{3/2}\)First, let's simplify the expression for \(x\). Recall that \(a^{m/n} = \sqrt[n]{a^m}\). So, \(2^{1/2} = \sqrt{2}\) and \(2^{3/2} = \sqrt{2^3} = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}\).
Substitute these simplified terms back into the expression for \(x\):
\(x = 2 + \sqrt{2} + 2\sqrt{2}\)Now, combine the terms with \(\sqrt{2}\):
\(x = 2 + (1+2)\sqrt{2}\) \(x = 2 + 3\sqrt{2}\)Simplifying the Expression \(x^2-4x-10\)
Our goal is to find the value of \(x^2-4x-10\). We can manipulate the simplified expression for \(x\) to find the value of \(x^2-4x\) more easily.
From \(x = 2 + 3\sqrt{2}\), subtract 2 from both sides:
\(x - 2 = 3\sqrt{2}\)Now, square both sides of the equation to eliminate the square root:
\((x - 2)^2 = (3\sqrt{2})^2\)Expand both sides:
\(x^2 - 2(x)(2) + 2^2 = 3^2 \times (\sqrt{2})^2\) \(x^2 - 4x + 4 = 9 \times 2\) \(x^2 - 4x + 4 = 18\)To find the value of \(x^2-4x\), subtract 4 from both sides:
\(x^2 - 4x = 18 - 4\) \(x^2 - 4x = 14\)Calculating the Final Value
Now we can substitute the value of \(x^2 - 4x\) into the expression we need to evaluate, which is \(x^2-4x-10\).
\(x^2 - 4x - 10 = (x^2 - 4x) - 10\)Substitute \(x^2 - 4x = 14\):
\(= 14 - 10\) \(= 4\)Therefore, the value of the expression \(x^2-4x-10\) is 4.
Similar Questions
- If \(\frac{1}{a}+\frac{1}{b}=\frac{5}{6}\) and \(\frac{1}{a^2}+\frac{1}{b^2}=\frac{13}{36}\), then what is \(\frac{1}{a^3}+\frac{1}{b^3}\) equal to?
- If \(x^4 = x^2 + 1\), where \(x > 0\), then what is \(2x^4\) equal to?
- If $\frac{\sqrt{p+x}+\sqrt{p-x}}{\sqrt{p+x}-\sqrt{p-x}} = p$, then what is $x$ equal to?
- If \(x-\frac{1}{x}=2\), \(x > 0\); then what is \(x^2-\frac{1}{x^2}\) equal to?
- If \((a-b)^2 + (b - c)^2 + (c-a)^2 = 6\) and \(a^2 + b^2+c^2 = 29\), then what is \((a + b + c)\) equal to?
- If \(p=\frac{\sqrt{5}+2}{\sqrt{5}-2}\) and \(q=\frac{\sqrt{5}-2}{\sqrt{5}+2}\), then what is \(\frac{p}{q}+\frac{q}{p}\) equal to?
Important Questions from Algebric Equations
A group of 630 children is seated in rows for a group photo session. Each row contains three less children than the row in front of it. Which one of the following number of rows is not possible?
The letters L, M, N, 0, P, Q, R, S and T in their order are substituted by nine integers 1 to 9 but not in that order. 4 is assigned to P. The difference between P and T is 5. The difference between N and T is 3.
What is the integer assigned to N?
Four persons, Alok, Bhupesh, Chander and Dinesh have a total of Rs. 100 among themselves. Alok and Bhupesh between them have as much money as Chander and Dinesh between them, but Alok has more money than Bhupesh; and Chander has only half the money that Dinesh has. Alok has in fact Rs. 5 more than Dinesh has.
Who has the maximum amount of money?
If x=3/2, then the value of 27x3-54x2+36x-11 is
If a+b+c = 6 and ab+bc+ca = 11, then the value of bc(b+c) + ca(c+a) +ab(a+b) +3abc is
