a) Determine the value of \( q \) if \( a=2, b=-1 \) and \( c=-4 \). Leave your naswer in simplest surd form. b) State whether \( q \) is rational or irational. c) Betweer which TWO consecutivo integers does \( q \) lie?

To address the problem, we'll assume that \( q \) is a root of the quadratic equation defined by the coefficients \( a = 2 \), \( b = -1 \), and \( c = -4 \). The general form of a quadratic equation is: \[ ax^2 + bx + c = 0 \] Substituting the given values: \[ 2q^2 - q - 4 = 0 \] ### a) Determining the Value of \( q \) in Simplest Surd Form To find the roots of the quadratic equation, we'll use the quadratic formula: \[ q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Plugging in the values: \[ q = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot (-4)}}{2 \cdot 2} \] \[ q = \frac{1 \pm \sqrt{1 + 32}}{4} \] \[ q = \frac{1 \pm \sqrt{33}}{4} \] Thus, the two possible values for \( q \) are: \[ q = \frac{1 + \sqrt{33}}{4} \quad \text{and} \quad q = \frac{1 - \sqrt{33}}{4} \] ### b) Nature of \( q \): Rational or Irrational The term \( \sqrt{33} \) is an irrational number because 33 is not a perfect square. Since \( q \) is expressed in terms of \( \sqrt{33} \), both possible values of \( q \) are **irrational**. ### c) Consecutive Integers Between Which \( q \) Lies Let's approximate the value of \( \sqrt{33} \): \[ \sqrt{25} = 5 \quad \text{and} \quad \sqrt{36} = 6 \] \[ 5 < \sqrt{33} < 6 \quad \Rightarrow \quad 1 + 5 < 1 + \sqrt{33} < 1 + 6 \quad \Rightarrow \quad 6 < 1 + \sqrt{33} < 7 \] \[ \frac{6}{4} = 1.5 \quad \text{and} \quad \frac{7}{4} = 1.75 \] \[ 1.5 < \frac{1 + \sqrt{33}}{4} < 1.75 \] Therefore, \( q \) lies **between the consecutive integers 1 and 2**. --- **Summary:** a) \( q = \dfrac{1 \pm \sqrt{33}}{4} \) in simplest surd form. b) \( q \) is **irrational**. c) \( q \) lies between the consecutive integers **1 and 2**.