Podcast
Questions and Answers
Solve for $x$ in the equation $2^{x+1} = 8^x$.
Solve for $x$ in the equation $2^{x+1} = 8^x$.
Start by noticing that $8 = 2^3$, so we can rewrite the equation as $2^{x+1} = (2^3)^x$. This becomes $2^{x+1} = 2^{3x}$, so $x+1=3x$, so $2x=1$, so $x=\frac{1}{2}$. Alternatively, we may write the equation as $2^{x+1} = 2^{3x}$ and since the bases are equal, the exponents are equal, so $x+1=3x$, so $2x=1$, so $x=\frac{1}{2}$. So, the solution is $x=\frac{1}{2}$.
Solve the inequality $x^2 - 4x - 3 > 0$.
Solve the inequality $x^2 - 4x - 3 > 0$.
Start by factorising the left-hand side of the inequality as $(x-3)(x+1) > 0$. The roots are $x = 3$ and $x = -1$, and the graph of the corresponding quadratic curve is above the $x$-axis for $x < -1$ and $x > 3. So, the solution is $x < -1$ or $x > 3.
Find the value of $x$ for which the quadratic expression $x^2 + 5x + 6$ has equal roots.
Find the value of $x$ for which the quadratic expression $x^2 + 5x + 6$ has equal roots.
Since the quadratic expression has equal roots, we know the discriminant is zero. Therefore, $(5)^2 - 4(1)(6) = 0$, so $25 - 24 = 0$, which is true, so it has equal roots. To find $x$, we use the quadratic formula: $x = \frac{-5 \pm \sqrt{0}}{2} = -\frac{5}{2}$. So, the solution is $x = -\frac{5}{2}.
The $n$th term of a quadratic sequence is given by $T_n = an^2 + bn + c$. If $T_1 = 5$, $T_2 = 12$, and $T_3 = 21$, find the values of $a$, $b$, and $c$.
The $n$th term of a quadratic sequence is given by $T_n = an^2 + bn + c$. If $T_1 = 5$, $T_2 = 12$, and $T_3 = 21$, find the values of $a$, $b$, and $c$.
Signup and view all the answers
Given that $f(x) = x^2 - 4x + 3$, prove that $f(x+1) = f(x) - 2x + 1.
Given that $f(x) = x^2 - 4x + 3$, prove that $f(x+1) = f(x) - 2x + 1.
Signup and view all the answers