Example Question - quadratic expressions
Here are examples of questions we've helped users solve.
Determining Coefficients in Quadratic Expressions
<p>For each quadratic expression, identify the coefficients a, b, and c using the standard form of a quadratic equation, which is \( ax^2 + bx + c \).</p> <p>(a) \( 2x^2 - 5x + 1 \): \( a = 2, b = -5, c = 1 \)</p> <p>(b) \( x^2 - 2x \): \( a = 1, b = -2, c = 0 \)</p> <p>(c) \( 2y^2 + 1 \): \( a = 2, b = 0, c = 1 \)</p> <p>(d) \( -\frac{1}{2}p^2 + 4p \): \( a = -\frac{1}{2}, b = 4, c = 0 \)</p> <p>(e) \( -x - 2x^2 \): \( a = -2, b = -1, c = 0 \)</p> <p>(f) \( 4x^2 \): \( a = 4, b = 0, c = 0 \)</p> <p>(g) \( h^2 + \frac{3}{2}h - 4 \): \( a = 1, b = \frac{3}{2}, c = -4 \)</p> <p>(h) \( \frac{1}{3}k^2 - 2 \): \( a = \frac{1}{3}, b = 0, c = -2 \)</p> <p>(i) \( 2(r - 3) \): expand to get \( 2r - 6 \): \( a = 2, b = 0, c = -6 \)</p>
Determine Values for Quadratic Expressions
<p>For each quadratic expression, we identify coefficients a, b, and c in the standard form \( ax^2 + bx + c \):</p> <p>(a) \( 2x^2 - 5x + 1 \): a = 2, b = -5, c = 1</p> <p>(b) \( x^2 - 2x \): a = 1, b = -2, c = 0</p> <p>(c) \( 2x^2 + 1 \): a = 2, b = 0, c = 1</p> <p>(d) \( -\frac{1}{2}p^2 + 4p \): a = -\frac{1}{2}, b = 4, c = 0</p> <p>(e) \( -1 - x - 2x^2 \): a = -2, b = -1, c = -1</p> <p>(f) \( 4x^2 \): a = 4, b = 0, c = 0</p> <p>(g) \( h^2 + \frac{3}{2}h - 4 \): a = 1, b = \frac{3}{2}, c = -4</p> <p>(h) \( \frac{1}{3}k^2 - 2 \): a = \frac{1}{3}, b = 0, c = -2</p> <p>(i) \( 2r(r - 3) \): a = 2, b = -6, c = 0</p>
Identifying Quadratic Expressions
<p>To determine if each expression is a quadratic expression, we must check if it can be written in the form of \( ax^2 + bx + c \), where \( a \neq 0 \).</p> <p>(a) \( x^2 - 5 \) is a quadratic expression (it has \( a = 1 \)).</p> <p>(b) \( 2x^2 - x \) is a quadratic expression (it has \( a = 2 \)).</p> <p>(c) \( 3y^2 - 3x + 1 \) is a quadratic expression (it has \( a = 3 \)).</p> <p>(d) \( 2m^2 \) is a quadratic expression (it has \( a = 2 \)).</p> <p>(e) \( \frac{p^2}{1} - \frac{1}{2}p + 3 \) is a quadratic expression (it has \( a = 1 \)).</p> <p>(f) \( n(n - 2) \) is not a quadratic expression in standard form (it needs expansion to determine the degree).</p>
Determine Coefficients of Quadratic Expressions
<p>a = 2, b = -5, c = 1</p> <p>a = 1, b = -2, c = 0</p> <p>a = -\frac{1}{2}, b = 4, c = 0</p> <p>a = -2, b = 0, c = 0</p> <p>a = 1, b = \frac{3}{2}, c = -4</p>
Inequality of Quadratic Expressions
La desigualdad mostrada en la imagen es \(1 + 2x^2 + 4y^2 > 0\). Para resolver esta desigualdad, notemos que \(2x^2\) y \(4y^2\) son siempre no negativos ya que cualquier número (positivo o negativo) elevado al cuadrado resultará en un número no negativo y estamos multiplicando por un coeficiente positivo. Como \(2x^2 \geq 0\) y \(4y^2 \geq 0\) para todos los valores de \(x\) y \(y\), entonces el valor mínimo que \(1 + 2x^2 + 4y^2\) puede tener es cuando tanto \(2x^2\) como \(4y^2\) son cero, eso ocurre cuando \(x = 0\) y \(y = 0\). Incluso en este caso, \(1 + 2x^2 + 4y^2 = 1 > 0\). Por lo tanto, la desigualdad \(1 + 2x^2 + 4y^2 > 0\) se cumple para todos los valores reales de \(x\) y \(y\). En términos de conjuntos, la solución a esta desigualdad sería el conjunto de todos los pares ordenados \((x, y)\) en el plano cartesiano, es decir, el conjunto de todos los puntos en el plano \(\mathbb{R}^2\).
Identifying Quadratic Expressions
A quadratic expression is an algebraic expression of the second degree, which means it contains at least one term that is squared. The general form of a quadratic expression is \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. We can examine each option to determine which one is not a quadratic expression: a) \( 3x^2 - x + 1 \): This is a quadratic expression with a leading term of degree 2 (\( 3x^2 \)). b) \( x^3 -4 \): This is not a quadratic expression because it has a term with a power of 3 (\( x^3 \)), which makes it a cubic expression. c) \( 4x^2 - 1 \): This is a quadratic expression with a leading term of degree 2 (\( 4x^2 \)). d) \( 9 - x^2 \): This is a quadratic expression, though written with terms in reverse order, with the squared term having a degree of 2 (\( -x^2 \)). So, the expression that is not a quadratic expression is option (b) \( x^3 -4 \).
Simplifying Expression with Exponents
To simplify the expression given in the image, we need to follow the laws of exponents for division and addition. The expression is: \[ \frac{10x^{n+4} + 4.125x^{n+2}}{3x^{n+3} - 20x^{n+1}} \] First, factor out the common terms involving \(x\) from the numerator and the denominator. 1. In the numerator, factor out the smallest power of \(x\) which is \(x^{n+2}\): \[ x^{n+2}(10x^{2} + 4.125) \] 2. In the denominator, factor out the smallest power of \(x\) which is \(x^{n+1}\): \[ x^{n+1}(3x^{2} - 20) \] Now we rewrite the expression: \[ \frac{x^{n+2}(10x^{2} + 4.125)}{x^{n+1}(3x^{2} - 20)} \] Next, cancel out the common \(x^{n+1}\) term: \[ \frac{x(10x^{2} + 4.125)}{3x^{2} - 20} \] Since \(x\) does not cancel anything further in the denominator, we leave the expression as is. The final simplified expression is: \[ \frac{x(10x^{2} + 4.125)}{3x^{2} - 20} \] Depending on the context or further instructions, there might be additional simplification possible, like factoring the quadratic expressions if they are factorable. However, without additional context, this is the simplified form based on the given expression.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com