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Questions and Answers
What is the value of $5(7 + 3)$?
What is the value of $5(7 + 3)$?
Simplify the expression $3x^2 - 2x - 5$.
Simplify the expression $3x^2 - 2x - 5$.
If $f(x) = 2x^3 - x^2 + 3$, what is $f'(x)$?
If $f(x) = 2x^3 - x^2 + 3$, what is $f'(x)$?
What is the sum of the roots of the equation $2x^2 - 5x + 3 = 0$?
What is the sum of the roots of the equation $2x^2 - 5x + 3 = 0$?
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If $f(x) = x^4 - 6x^2 + 9$, what is the value of $f''(x)$?
If $f(x) = x^4 - 6x^2 + 9$, what is the value of $f''(x)$?
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Solve the inequality $2x^2 - 8x + 7 < 0$ for $x$.
Solve the inequality $2x^2 - 8x + 7 < 0$ for $x$.
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Study Notes
Algebraic Expressions and Calculations
- To evaluate (5(7 + 3)), first simplify the expression inside the parentheses: (7 + 3 = 10). Thus, (5 \times 10 = 50).
Simplification
- The expression (3x^2 - 2x - 5) is already in its simplest form as a quadratic expression.
Derivatives
- The derivative of the function (f(x) = 2x^3 - x^2 + 3) is computed as follows:
- (f'(x) = 6x^2 - 2x).
Roots of Quadratic Equations
- For the equation (2x^2 - 5x + 3 = 0), use the formula for the sum of the roots given by (-\frac{b}{a}):
- Here, (a = 2) and (b = -5), so the sum of the roots is (\frac{5}{2}).
Higher Order Derivatives
- To find the second derivative (f''(x)) of the function (f(x) = x^4 - 6x^2 + 9):
- First, calculate the first derivative: (f'(x) = 4x^3 - 12x).
- Then, find the second derivative: (f''(x) = 12x^2 - 12).
Solving Inequalities
- For the inequality (2x^2 - 8x + 7 < 0), rewrite it as (x^2 - 4x + \frac{7}{2} < 0).
- Find the roots using the quadratic formula (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}):
- Here, (a = 2), (b = -8), and (c = 7).
- The expression may take negative values between the roots if they exist within the interval determined by the quadratic's graph.
- Find the roots using the quadratic formula (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}):
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Description
Test your math skills with this IGCSE level exam question. Solve arithmetic and algebraic expressions, and find the derivative of a given function.
