GCSE Maths Formulas

\( |x| = x \text{ if } x \ge 0,\; -x \text{ if } x < 0 \)

\( \frac{A}{B}=\frac{C}{D}\;(BD\ne0)\;\Rightarrow\; AD=BC \)

\( u_n = a + (n-1)d \)

\( \text{Average rate on }[x_1,x_2]=\frac{f(x_2)-f(x_1)}{x_2-x_1} \)

\( (a+b)^2=a^2+2ab+b^2,\qquad (a-b)^2=a^2-2ab+b^2 \)

\( x^2+bx+c=\left(x+\tfrac{b}{2}\right)^2-\left(\tfrac{b^2}{4}-c\right) \)

\( a^2-b^2=(a-b)(a+b) \)

\( \Delta = b^2 - 4ac \)

\( x^2+bx+c=(x+p)(x+q),\;p+q=b,\;pq=c \)

\( a^{1/n}=\sqrt[n]{a},\qquad a^{m/n}=\sqrt[n]{a^{m}}\;(a>0) \)

\( u_n = ar^{\,n-1} \)

\( a^m a^n = a^{m+n},\quad \frac{a^m}{a^n}=a^{m-n},\quad (a^m)^n=a^{mn},\quad a^{-n}=\frac{1}{a^n},\quad a^0=1 \)

\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

\( u_n=an^2+bn+c,\quad a=\tfrac{1}{2}\,\Delta^2,\quad b=\Delta u_1-3a,\quad c=u_1-a-b \)

\( x_{\text{vertex}}= -\tfrac{b}{2a},\quad y_{\text{vertex}}= f\!\left(-\tfrac{b}{2a}\right),\quad \text{axis}:\;x=-\tfrac{b}{2a} \)

\( y=a(x-h)^2+k \)

\( y=a(x-r_1)(x-r_2)\;\Rightarrow\; r_1+r_2=-\tfrac{b}{a},\; r_1r_2=\tfrac{c}{a} \)

\( \frac{1}{\sqrt{a}+\sqrt{b}}\cdot\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}-\sqrt{b}}=\frac{\sqrt{a}-\sqrt{b}}{a-b} \)

\( \frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a}\;(a>0) \)

\( \frac{1}{a+\sqrt{b}}\times\frac{a-\sqrt{b}}{a-\sqrt{b}}=\frac{a-\sqrt{b}}{a^{2}-b} \)

\( ax+b=c\;\Rightarrow\;x=\tfrac{c-b}{a}\;(a\ne0) \)

\( S_n=\tfrac{n}{2}\,[2a+(n-1)d]=\tfrac{n}{2}(a_1+a_n) \)

\( \sqrt{ab}=\sqrt{a}\,\sqrt{b},\quad \sqrt{\tfrac{a}{b}}=\tfrac{\sqrt{a}}{\sqrt{b}}\;(b>0),\quad (\sqrt{a})^2=a \)

\( a\sqrt{b}+c\sqrt{b}=(a+c)\sqrt{b} \)