Solve for x

x\in \begin{bmatrix}3,4\end{bmatrix}

$x∈[3,4 ]$

Graph

Copy

Copied to clipboard

x^{2}-7x+12=0

To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 1\times 12}}{2}

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -7 for b, and 12 for c in the quadratic formula.

x=\frac{7±1}{2}

Do the calculations.

x=4 x=3

Solve the equation x=\frac{7±1}{2} when ± is plus and when ± is minus.

\left(x-4\right)\left(x-3\right)\leq 0

Rewrite the inequality by using the obtained solutions.

x-4\geq 0 x-3\leq 0

For the product to be ≤0, one of the values x-4 and x-3 has to be ≥0 and the other has to be ≤0. Consider the case when x-4\geq 0 and x-3\leq 0.

x\in \emptyset

This is false for any x.

x-3\geq 0 x-4\leq 0

Consider the case when x-4\leq 0 and x-3\geq 0.

x\in \begin{bmatrix}3,4\end{bmatrix}

The solution satisfying both inequalities is x\in \left[3,4\right].

x\in \begin{bmatrix}3,4\end{bmatrix}

The final solution is the union of the obtained solutions.

Examples

Quadratic equation

{ x } ^ { 2 } - 4 x - 5 = 0

$x_{2}−4x−5=0$

Trigonometry

4 \sin \theta \cos \theta = 2 \sin \theta

$4sinθcosθ=2sinθ$

Linear equation

y = 3x + 4

$y=3x+4$

Arithmetic

699 * 533

$699∗533$

Matrix

\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]

$[25 34 ][2−1 01 35 ]$

Simultaneous equation

\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.

${8x+2y=467x+3y=47 $

Differentiation

\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }

$dxd (x−5)(3x_{2}−2) $

Integration

\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x

$∫_{0}xe_{−x_{2}}dx$

Limits

\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}

$x→−3lim x_{2}+2x−3x_{2}−9 $